Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the substituted variable. In advanced trigonometry, these formulas are essential tools used to evaluate non-standard angles, simplify complex expressions, and prove other mathematical properties.

Sum and Difference Formulas

Sum and difference formulas allow you to expand the sine, cosine, or tangent of a sum or difference of two angles ($\alpha$ and $\beta$).

Sine: $\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$

Cosine: $\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$ (Note the sign change: a plus inside the cosine becomes a minus in the expansion, and vice versa.)

Tangent: $\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$

Example: Find the exact value of $\cos(75^\circ)$. We can split $75^\circ$ into two standard angles: $45^\circ + 30^\circ$. $\cos(75^\circ) = \cos(45^\circ + 30^\circ)$ $\cos(45^\circ + 30^\circ) = \cos(45^\circ)\cos(30^\circ) - \sin(45^\circ)\sin(30^\circ)$ $= \left(\frac{\sqrt{2}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right)\left(\frac{1}{2}\right) = \frac{\sqrt{6} - \sqrt{2}}{4}$

Double-Angle Formulas

Double-angle formulas are derived directly from the sum formulas by setting $\alpha = \beta = \theta$. They are incredibly useful for simplifying expressions where an angle is multiplied by 2.

Sine: $\sin(2\theta) = 2\sin\theta\cos\theta$

Cosine: (This has three variations depending on what is most convenient)

1. $\cos(2\theta) = \cos^2\theta - \sin^2\theta$
2. $\cos(2\theta) = 2\cos^2\theta - 1$
3. $\cos(2\theta) = 1 - 2\sin^2\theta$

Tangent: $\tan(2\theta) = \frac{2\tan\theta}{1 - \tan^2\theta}$

Half-Angle Formulas

Half-angle formulas are derived from the double-angle formulas for cosine. They help find the trigonometric values of half of a known angle. The $\pm$ sign is determined by the quadrant in which the angle $\frac{\theta}{2}$ lies.

Sine: $\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$

Cosine: $\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$

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

Proving Identities

Proving a trigonometric identity means showing that one side of the equation is identical to the other. The best strategy is usually to start with the more complex side and use known formulas to simplify it until it matches the simpler side.

Example: Prove: $\frac{1 + \cos 2\theta}{\sin 2\theta} = \cot \theta$

Let's start with the Left Hand Side (LHS): $\text{LHS} = \frac{1 + \cos 2\theta}{\sin 2\theta}$

Substitute the double-angle formulas. For the numerator, using $\cos(2\theta) = 2\cos^2\theta - 1$ is a smart choice because it will cancel out the $1$: $\text{LHS} = \frac{1 + (2\cos^2\theta - 1)}{2\sin\theta\cos\theta}$

Simplify the numerator: $\text{LHS} = \frac{2\cos^2\theta}{2\sin\theta\cos\theta}$

Cancel the common factors ($2$ and one $\cos\theta$): $\text{LHS} = \frac{\cos\theta}{\sin\theta}$

By the quotient identity, this equals cotangent: $\text{LHS} = \cot\theta = \text{RHS}$

The identity is successfully proven.