• Apply relevant identities to rewrite terms:
• Pythagorean identities: $\sin^2 \theta + \cos^2 \theta = 1$ $1 + \tan^2 \theta = \sec^2 \theta$ $\csc^2 \theta = 1 + \cot^2 \theta$
• Reciprocal identities: $\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}$
• Quotient identities: $\tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}$
• Double angle identities (if applicable): $\sin 2\theta = 2\sin \theta \cos \theta$ $\cos 2\theta = \cos^2 \theta - \sin^2 \theta$
• Sum-to-product or product-to-sum identities (if applicable).

Example 1: Prove that $\ \sin^2 \theta + \cos^2 \theta = 1$.

• Solution:
• This is a fundamental identity in trigonometry.
• Start by recognizing that the equation is true by definition, as it's derived from the Pythagorean theorem in a right triangle where $\ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$ and $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$.
• Hence: $\sin^2 \theta + \cos^2 \theta = \frac{\text{opposite}^2}{\text{hypotenuse}^2} + \frac{\text{adjacent}^2}{\text{hypotenuse}^2} = \frac{\text{opposite}^2 + \text{adjacent}^2}{\text{hypotenuse}^2} = \frac{\text{hypotenuse}^2}{\text{hypotenuse}^2} = 1$

Example 2: Prove that $\ \sec^2 \theta - \tan^2 \theta = 1$.

• Solution:
• Start with the left-hand side (LHS): $\sec^2 \theta - \tan^2 \theta$
• Use the identities $\ \sec \theta = \frac{1}{\cos \theta} \ and \ \tan \theta = \frac{\sin \theta}{\cos \theta}$ to rewrite the expression: $\sec^2 \theta = 1 + \tan^2 \theta$
• Substitute into the LHS: $\sec^2 \theta - \tan^2 \theta = (1 + \tan^2 \theta) - \tan^2 \theta = 1$
• Thus, LHS = RHS, and the identity is proven.

Example 3: Prove that $\ \sin(2\theta) = 2\sin(\theta)\cos(\theta) .$

• Solution:
• Use the sum identity for sine: $\sin(2\theta) = \sin(\theta + \theta)$
• Apply the sine sum identity: $\sin(\theta + \theta) = \sin \theta \cos \theta + \cos \theta \sin \theta$
• Combine like terms: $\sin(2\theta) = 2\sin \theta \cos \theta$
• Thus, the identity is proven.

• Trigonometric proofs require a strategic approach, focusing on simplifying one or both sides of the identity until they match.
• The use of fundamental trigonometric identities, algebraic manipulation, and pattern recognition are key to successfully proving trigonometric identities.
• Mastery of these techniques allows you to tackle a wide range of trigonometric problems with confidence.