Trigonometric Identities Problems

[MathDefinition title="Trigonometric Identities"] Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables where both sides are defined. The three fundamental identities are:

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = cosec²θ [/MathDefinition]

[MathDefinition title="Quotient Identities"]

1. tan θ = sin θ / cos θ
2. cot θ = cos θ / sin θ [/MathDefinition]

[MathKeyFormula title="Key Identities to Remember"]

1. sin²θ + cos²θ = 1
2. 1 + tan²θ = sec²θ
3. 1 + cot²θ = cosec²θ
4. sin 2θ = 2 sin θ cos θ
5. cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
6. sin(A ± B) = sin A cos B ± cos A sin B
7. cos(A ± B) = cos A cos B ∓ sin A sin B [/MathKeyFormula]

[MathNote title="Strategy for Proving Identities"]

1. Start with the more complex side
2. Convert all functions to sine and cosine if needed
3. Use fundamental identities to simplify
4. Factor or combine terms as needed
5. Work toward the simpler side [/MathNote]

[MathExample title="Example 1: Basic Identity Proof"] Problem: Prove that (1 − sin² A) = cos² A.

Solution: LHS = 1 − sin² A

Using the identity sin² A + cos² A = 1: 1 − sin² A = cos² A

Therefore, LHS = cos² A = RHS

Hence proved. [/MathExample]

[MathExample title="Example 2: Proving with Multiple Steps"] Problem: Prove that (sin A + cos A)² = 1 + 2 sin A cos A.

Solution: LHS = (sin A + cos A)² = sin² A + 2 sin A cos A + cos² A = (sin² A + cos² A) + 2 sin A cos A = 1 + 2 sin A cos A = RHS

Hence proved. [/MathExample]

[MathExample title="Example 3: Complex Identity"] Problem: Prove that (1 + tan² A) / (1 + cot² A) = tan² A.

Solution: LHS = (1 + tan² A) / (1 + cot² A)

Using identities: 1 + tan² A = sec² A and 1 + cot² A = cosec² A

LHS = sec² A / cosec² A = (1/cos² A) / (1/sin² A) = sin² A / cos² A = tan² A = RHS

Hence proved. [/MathExample]

[MathExample title="Example 4: Proving with Substitution"] Problem: Prove that sin⁴ A − cos⁴ A = sin² A − cos² A.

Solution: LHS = sin⁴ A − cos⁴ A = (sin² A)² − (cos² A)² = (sin² A − cos² A)(sin² A + cos² A) [Using a² − b² = (a − b)(a + b)] = (sin² A − cos² A)(1) [Since sin² A + cos² A = 1] = sin² A − cos² A = RHS

Hence proved. [/MathExample]