Fundamental Trigonometric Identities

Introduction to Trigonometric Identities

A trigonometric identity is an equation involving trigonometric ratios of an angle (or angles) that is true for all permissible values of the variable(s) for which both sides of the equation are defined. In simpler terms, it's a statement of equality between two trigonometric expressions that holds true universally, whenever the expressions make sense.

The "permissible values" refer to the domain of the trigonometric functions involved. For example, identities involving $\tan \theta$ or $\sec \theta$ are not defined for angles like $90^\circ, 270^\circ$, etc. where $\cos \theta = 0$. Similarly, identities involving $\cot \theta$ or $\text{cosec} \, \theta$ are not defined for angles like $0^\circ, 180^\circ$, etc. where $\sin \theta = 0$. Identities hold true for all other values.

Trigonometric identities are fundamental tools in mathematics and its applications in physics, engineering, and other fields. They are used extensively to:

• Simplify trigonometric expressions: Allowing complex expressions to be rewritten in simpler forms.

• Solve trigonometric equations: Transforming equations into solvable forms.

• Prove other identities or theorems: Serving as foundational relationships.

• Perform calculations in calculus: For integrating or differentiating trigonometric functions.

There are many trigonometric identities, but the most basic and frequently used ones are the Reciprocal Identities, Quotient Identities, and the Pythagorean Identities.

Pythagorean Identities

These three fundamental trigonometric identities are derived directly from the Pythagorean Theorem, which states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

Consider a right-angled triangle ABC, with the right angle at B. Let $\angle C = \theta$.

With respect to angle $\theta$:

• Opposite Side (O) = AB

• Adjacent Side (A) = BC

• Hypotenuse (H) = AC

According to the Pythagorean Theorem:

Or, using the labels O, A, H:

We can derive three key identities by dividing this equation by $H^2$, $A^2$, or $O^2$.

1. Identity: $\sin^2 \theta + \cos^2 \theta = 1$

Derivation:

Start with the Pythagorean Theorem (Equation 1):

Divide every term in Equation (1) by the square of the Hypotenuse, $H^2$. Note that $H \ne 0$ for any triangle.

$ \frac{O^2}{H^2} + \frac{A^2}{H^2} = \frac{H^2}{H^2} $

Rewrite the terms using exponents:

$ \left(\frac{O}{H}\right)^2 + \left(\frac{A}{H}\right)^2 = \left(\frac{H}{H}\right)^2 $

Recall the definitions of sine and cosine in a right triangle:

• $ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{O}{H} $

• $ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{A}{H} $

Also, $ \frac{H}{H} = 1 $.

Substitute these into the equation:

$ (\sin \theta)^2 + (\cos \theta)^2 = (1)^2 $

Using the standard notation $\sin^2 \theta$ for $(\sin \theta)^2$ and $\cos^2 \theta$ for $(\cos \theta)^2$:

$\mathbf{\sin^2 \theta + \cos^2 \theta = 1}$

This identity is valid for all real values of $\theta$, as sine and cosine are defined for all angles.

Variations of this identity:

• $ \sin^2 \theta = 1 - \cos^2 \theta $

• $ \cos^2 \theta = 1 - \sin^2 \theta $

2. Identity: $1 + \tan^2 \theta = \sec^2 \theta$

Derivation:

Start again with the Pythagorean Theorem (Equation 1):

Divide every term in Equation (1) by the square of the Adjacent side, $A^2$. This requires $A \ne 0$, which means the angle $\theta$ cannot be $90^\circ, 270^\circ, \dots$, i.e., $\cos \theta \ne 0$.

$ \frac{O^2}{A^2} + \frac{A^2}{A^2} = \frac{H^2}{A^2} $

Rewrite the terms using exponents:

$ \left(\frac{O}{A}\right)^2 + \left(\frac{A}{A}\right)^2 = \left(\frac{H}{A}\right)^2 $

Recall the definitions of tangent and secant:

• $ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{O}{A} $

• $ \sec \theta = \frac{\text{Hypotenuse}}{\text{Adjacent}} = \frac{H}{A} $

Also, $ \frac{A}{A} = 1 $.

Substitute these into the equation:

$ (\tan \theta)^2 + (1)^2 = (\sec \theta)^2 $

Using standard notation:

$\mathbf{\tan^2 \theta + 1 = \sec^2 \theta}$ or $\mathbf{1 + \tan^2 \theta = \sec^2 \theta}$

This identity is valid for all values of $\theta$ for which $\tan \theta$ and $\sec \theta$ are defined (i.e., $\theta \ne (2n+1)90^\circ$ for integer n).

Variations of this identity:

• $ \sec^2 \theta - 1 = \tan^2 \theta $

• $ \sec^2 \theta - \tan^2 \theta = 1 $ (useful for factorising difference of squares: $(\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1$)

3. Identity: $1 + \cot^2 \theta = \text{cosec}^2 \theta$

Derivation:

Start with the Pythagorean Theorem (Equation 1):

Divide every term in Equation (1) by the square of the Opposite side, $O^2$. This requires $O \ne 0$, which means the angle $\theta$ cannot be $0^\circ, 180^\circ, 360^\circ, \dots$, i.e., $\sin \theta \ne 0$.

$ \frac{O^2}{O^2} + \frac{A^2}{O^2} = \frac{H^2}{O^2} $

Rewrite the terms using exponents:

$ \left(\frac{O}{O}\right)^2 + \left(\frac{A}{O}\right)^2 = \left(\frac{H}{O}\right)^2 $

Recall the definitions of cotangent and cosecant:

• $ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = \frac{A}{O} $

• $ \text{cosec} \, \theta = \frac{\text{Hypotenuse}}{\text{Opposite}} = \frac{H}{O} $

Also, $ \frac{O}{O} = 1 $.

Substitute these into the equation:

$ (1)^2 + (\cot \theta)^2 = (\text{cosec} \, \theta)^2 $

Using standard notation:

$\mathbf{1 + \cot^2 \theta = \text{cosec}^2 \theta}$

This identity is valid for all values of $\theta$ for which $\cot \theta$ and $\text{cosec} \, \theta$ are defined (i.e., $\theta \ne n \times 180^\circ$ for integer n).

Variations of this identity:

• $ \text{cosec}^2 \theta - 1 = \cot^2 \theta $

• $ \text{cosec}^2 \theta - \cot^2 \theta = 1 $ (useful for factorising difference of squares: $(\text{cosec} \, \theta - \cot \theta)(\text{cosec} \, \theta + \cot \theta) = 1$)

These three Pythagorean identities ($\sin^2 \theta + \cos^2 \theta = 1$, $1 + \tan^2 \theta = \sec^2 \theta$, and $1 + \cot^2 \theta = \text{cosec}^2 \theta$) are fundamental and must be memorised. They are the basis for proving many other trigonometric identities.

Quotient and Reciprocal Identities

In addition to the Pythagorean identities, there are other fundamental identities that are derived directly from the definitions of the trigonometric ratios in a right triangle. These include the Reciprocal Identities and the Quotient Identities.

Reciprocal Identities

These identities express the relationship between pairs of trigonometric ratios where one is the reciprocal of the other. They follow directly from the definitions based on Opposite, Adjacent, and Hypotenuse sides (O, A, H).

• Sine and Cosecant:

  $ \sin \theta = \frac{O}{H} $ and $ \text{cosec} \, \theta = \frac{H}{O} $

  Since $\text{cosec} \, \theta$ is the ratio $\frac{H}{O}$, which is the reciprocal of $\frac{O}{H}$, we have:

  $\mathbf{\text{cosec} \, \theta = \frac{1}{\sin \theta}}$

  This identity is valid for all $\theta$ where $\sin \theta \ne 0$ (i.e., $\theta \ne n \times 180^\circ$). It also implies $\sin \theta = \frac{1}{\text{cosec} \, \theta}$ and $ \sin \theta \cdot \text{cosec} \, \theta = 1 $.

• Cosine and Secant:

  $ \cos \theta = \frac{A}{H} $ and $ \sec \theta = \frac{H}{A} $

  Since $\sec \theta$ is the ratio $\frac{H}{A}$, which is the reciprocal of $\frac{A}{H}$, we have:

  $\mathbf{\sec \theta = \frac{1}{\cos \theta}}$

  This identity is valid for all $\theta$ where $\cos \theta \ne 0$ (i.e., $\theta \ne (2n+1)90^\circ$). It also implies $\cos \theta = \frac{1}{\sec \theta}$ and $ \cos \theta \cdot \sec \theta = 1 $.

• Tangent and Cotangent:

  $ \tan \theta = \frac{O}{A} $ and $ \cot \theta = \frac{A}{O} $

  Since $\cot \theta$ is the ratio $\frac{A}{O}$, which is the reciprocal of $\frac{O}{A}$, we have:

  $\mathbf{\cot \theta = \frac{1}{\tan \theta}}$

  This identity is valid for all $\theta$ where $\tan \theta$ is defined and non-zero (i.e., $\theta \ne n \times 90^\circ$). It also implies $\tan \theta = \frac{1}{\cot \theta}$ and $ \tan \theta \cdot \cot \theta = 1 $.

Quotient Identities

These identities express the tangent and cotangent ratios in terms of the sine and cosine ratios.

• Tangent in terms of Sine and Cosine:

  We know that $ \tan \theta = \frac{O}{A} $. We can express O and A in terms of H using sine and cosine definitions:

  From $ \sin \theta = \frac{O}{H} $, we get $ O = H \sin \theta $.

  From $ \cos \theta = \frac{A}{H} $, we get $ A = H \cos \theta $.

  Now substitute these into the expression for $ \tan \theta $:

  $ \tan \theta = \frac{O}{A} = \frac{H \sin \theta}{H \cos \theta} $

  Assuming $H \ne 0$ and $\cos \theta \ne 0$, we can cancel H from the numerator and denominator:

  $ \tan \theta = \frac{\cancel{H} \sin \theta}{\cancel{H} \cos \theta} = \frac{\sin \theta}{\cos \theta} $

  Thus, the quotient identity for tangent is:

  $\mathbf{\tan \theta = \frac{\sin \theta}{\cos \theta}}$

  This identity is valid for all $\theta$ where $\cos \theta \ne 0$ (i.e., $\theta \ne (2n+1)90^\circ$).

• Cotangent in terms of Sine and Cosine:

  We know that $ \cot \theta = \frac{A}{O} $. Using $O = H \sin \theta$ and $A = H \cos \theta$:

  $ \cot \theta = \frac{A}{O} = \frac{H \cos \theta}{H \sin \theta} $

  Assuming $H \ne 0$ and $\sin \theta \ne 0$, we can cancel H:

  $ \cot \theta = \frac{\cancel{H} \cos \theta}{\cancel{H} \sin \theta} = \frac{\cos \theta}{\sin \theta} $

  Thus, the quotient identity for cotangent is:

  $\mathbf{\cot \theta = \frac{\cos \theta}{\sin \theta}}$

  This identity is valid for all $\theta$ where $\sin \theta \ne 0$ (i.e., $\theta \ne n \times 180^\circ$).

Note that the quotient identity for $\cot \theta$ can also be obtained directly from the reciprocal identity $\cot \theta = \frac{1}{\tan \theta}$ and the quotient identity for $\tan \theta$:

$ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\left(\frac{\sin \theta}{\cos \theta}\right)} = 1 \times \frac{\cos \theta}{\sin \theta} = \frac{\cos \theta}{\sin \theta} $

These Reciprocal and Quotient identities, along with the Pythagorean identities, form the core set of fundamental trigonometric identities. Mastery of these is crucial before moving on to more complex identities and applications.

Using Identities to Simplify Expressions and Prove Identities

Trigonometric identities are powerful tools that allow us to manipulate trigonometric expressions and equations. They serve a similar purpose to algebraic identities (like $(a+b)^2 = a^2 + 2ab + b^2$) but apply specifically to trigonometric ratios. The fundamental identities we have discussed (Reciprocal, Quotient, and Pythagorean Identities) are the building blocks for simplifying more complex expressions and proving other identities.

1. Simplifying Trigonometric Expressions

Simplifying a trigonometric expression means rewriting it in a more concise, understandable, or usable form. There is often no single "most simple" form, as the goal depends on the context (e.g., preparing for differentiation, integration, or solving an equation). However, common simplification goals include reducing the number of terms, expressing the result in terms of fewer trigonometric functions (often sine and cosine), or eliminating fractions or square roots.

General Strategies for Simplifying:

• Convert to Sine and Cosine: A very common and often effective strategy is to express all tangent, cotangent, secant, and cosecant terms in terms of sine and cosine using the reciprocal and quotient identities. This reduces the number of different functions you are dealing with.

• Look for Pythagorean Identities: Identify terms like $\sin^2 \theta + \cos^2 \theta$ (which equals 1), $\sec^2 \theta - 1$ (which equals $\tan^2 \theta$), $\text{cosec}^2 \theta - 1$ (which equals $\cot^2 \theta$), etc., and make the appropriate substitutions.

• Apply Algebraic Techniques: Use standard algebraic methods such as:

  • Factoring (e.g., difference of squares, perfect squares, common factors)

  • Combining fractions by finding a common denominator

  • Multiplying by a conjugate (especially with expressions involving $1 \pm \sin \theta$ or $1 \pm \cos \theta$)

  • Expanding expressions (though often the goal is to avoid this)

• Cancel Common Factors: After factoring or manipulating, look for terms in the numerator and denominator that can be cancelled.

• Work Systematically: Don't try to do too many steps at once. Simplify one part of the expression at a time.

Example 1: Simplifying a Trigonometric Expression

Answer:

2. Proving Trigonometric Identities

Proving a trigonometric identity means demonstrating that one side of the equation can be transformed into the other side using known identities and valid algebraic steps, for all permissible values of the variable(s).

General Strategies for Proving:

1. Work on One Side: Start with the side of the identity that appears more complex and try to simplify it to match the other side. It is generally easier to simplify a complicated expression than to complicate a simple one.

2. Convert to Sine and Cosine: As in simplification, converting all terms to sine and cosine is often a helpful first step, especially if both sides seem equally complex or involve a mix of different trigonometric functions.

3. Look for Algebraic Opportunities: See if you can apply factoring, combining fractions, splitting fractions, multiplying by a conjugate, etc.

4. Look for Identity Substitutions: Keep the fundamental identities in mind and look for opportunities to substitute expressions (e.g., replace $\tan^2 \theta + 1$ with $\sec^2 \theta$, or $1$ with $\sin^2 \theta + \cos^2 \theta$).

5. Manipulate Both Sides (Less Common): If you get stuck working on one side, you can sometimes work on both sides independently, transforming each into the same third expression.

6. Do NOT Assume the Identity is True: Do not perform operations that involve moving terms across the equality sign (e.g., adding or subtracting the same term from both sides), as this assumes the equality is already true, which is what you are trying to prove. Work only on one side at a time or manipulate both sides independently.

7. Be Clear and Show All Steps: Each step should be justifiable by an identity or an algebraic rule. Clearly show how you get from one step to the next.

Example 2: Proving a Trigonometric Identity

Answer:

Note for Competitive Exams