Understanding Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where the functions are defined. These identities are useful in simplifying expressions and solving trigonometric equations. They are essential tools in various areas of mathematics, including geometry, calculus, and complex number theory.

Fundamental Trigonometric Identities

There are several fundamental trigonometric identities that are the basis for more complex identities. Here are the most important ones:

Pythagorean Identities

These identities are derived from the Pythagorean theorem and relate the squares of the sine and cosine functions.

1. $\sin^2\theta + \cos^2\theta = 1$
2. $1 + \tan^2\theta = \sec^2\theta$
3. $1 + \cot^2\theta = \csc^2\theta$

Reciprocal Identities

These identities show the relationship between the basic trigonometric functions and their reciprocals.

1. $\sin\theta = \frac{1}{\csc\theta}$
2. $\cos\theta = \frac{1}{\sec\theta}$
3. $\tan\theta = \frac{1}{\cot\theta}$
4. $\csc\theta = \frac{1}{\sin\theta}$
5. $\sec\theta = \frac{1}{\cos\theta}$
6. $\cot\theta = \frac{1}{\tan\theta}$

Quotient Identities

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

1. $\tan\theta = \frac{\sin\theta}{\cos\theta}$
2. $\cot\theta = \frac{\cos\theta}{\sin\theta}$

Co-Function Identities

These identities relate the trigonometric functions of complementary angles.

1. $\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$
2. $\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$
3. $\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$
4. $\cot\left(\frac{\pi}{2} - \theta\right) = \tan\theta$
5. $\sec\left(\frac{\pi}{2} - \theta\right) = \csc\theta$
6. $\csc\left(\frac{\pi}{2} - \theta\right) = \sec\theta$

Even-Odd Identities

Trigonometric functions have properties that determine whether they are even or odd functions.

1. $\sin(-\theta) = -\sin\theta$ (odd function)
2. $\cos(-\theta) = \cos\theta$ (even function)
3. $\tan(-\theta) = -\tan\theta$ (odd function)
4. $\cot(-\theta) = -\cot\theta$ (odd function)
5. $\sec(-\theta) = \sec\theta$ (even function)
6. $\csc(-\theta) = -\csc\theta$ (odd function)

Sum and Difference Identities

These identities express the sine, cosine, and tangent of the sum or difference of two angles.

1. $\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta$
2. $\cos(\alpha \pm \beta) = \cos\alpha \cos\beta \mp \sin\alpha \sin\beta$
3. $\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha \tan\beta}$

Double Angle Identities

These identities are used to express functions of double angles in terms of single angles.

1. $\sin(2\theta) = 2\sin\theta\cos\theta$
2. $\cos(2\theta) = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
3. $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Half-Angle Identities

These identities are used to express functions of half angles in terms of the square root of expressions involving the original angle.

1. $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$
2. $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$
3. $\tan\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{1 + \cos\theta}} = \frac{\sin\theta}{1 + \cos\theta} = \frac{1 - \cos\theta}{\sin\theta}$

Product-to-Sum and Sum-to-Product Identities

These identities are used to convert products of trigonometric functions into sums or differences and vice versa.

1. $\sin\alpha \cos\beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)]$
2. $\cos\alpha \sin\beta = \frac{1}{2}[\sin(\alpha + \beta) - \sin(\alpha - \beta)]$
3. $\cos\alpha \cos\beta = \frac{1}{2}[\cos(\alpha + \beta) + \cos(\alpha - \beta)]$
4. $\sin\alpha \sin\beta = \frac{1}{2}[\cos(\alpha - \beta) - \cos(\alpha + \beta)]$

Examples

To illustrate the use of these identities, let's look at a couple of examples:

Example 1: Simplifying an Expression

Simplify the expression $\sin^2\theta \cdot \tan^2\theta$.

Using the Pythagorean identity $\tan^2\theta = \sec^2\theta - 1$ and the reciprocal identity $\sec\theta = \frac{1}{\cos\theta}$, we get:

$$ \sin^2\theta \cdot \tan^2\theta = \sin^2\theta \cdot (\sec^2\theta - 1) = \sin^2\theta \cdot \left(\frac{1}{\cos^2\theta} - 1\right) = \frac{\sin^2\theta}{\cos^2\theta} - \sin^2\theta = \tan^2\theta - \sin^2\theta $$

Example 2: Finding the Exact Value

Find the exact value of $\cos\left(\frac{\pi}{4} - \frac{\pi}{6}\right)$.

Using the sum and difference identity for cosine:

$$ \cos\left(\frac{\pi}{4} - \frac{\pi}{6}\right) = \cos\frac{\pi}{4}\cos\frac{\pi}{6} + \sin\frac{\pi}{4}\sin\frac{\pi}{6} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6} + \sqrt{2}}{4} $$

Table of Differences and Important Points

Identity Type	Example	Important Point
Pythagorean	$\sin^2\theta + \cos^2\theta = 1$	Relates squares of sine and cosine
Reciprocal	$\sin\theta = \frac{1}{\csc\theta}$	Expresses functions in terms of their reciprocals
Quotient	$\tan\theta = \frac{\sin\theta}{\cos\theta}$	Defines tangent and cotangent in terms of sine and cosine
Co-Function	$\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$	Relates functions of complementary angles
Even-Odd	$\sin(-\theta) = -\sin\theta$	Determines symmetry of functions
Sum and Difference	$\sin(\alpha \pm \beta) = \sin\alpha \cos\beta \pm \cos\alpha \sin\beta$	Expresses functions of angle sums/differences
Double Angle	$\sin(2\theta) = 2\sin\theta\cos\theta$	Simplifies expressions involving double angles
Half-Angle	$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$	Useful for integration and solving equations
Product-to-Sum	$\sin\alpha \cos\beta = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)]$	Converts products to sums/differences

Understanding and applying these identities is crucial for solving trigonometric problems and simplifying complex expressions. Practice using these identities in various contexts to become proficient in their application.