Question

Prove the following trigonometric identities:
(1+sinθ-cosθ/1+sinθ+cosθ)²=1-cosθ/1+cosθ

Answer 1

Answer:

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Answer 2

(1+sinθ-cosθ / 1+sinθ+cosθ)² = 1-cosθ / 1+cosθ proved

Step-by-step explanation:

 To prove: (1+sinθ-cosθ / 1+sinθ+cosθ)² = 1-cosθ / 1+cosθ

Proof: Sin²θ = 1 - Cos²θ

Sinθ. Sinθ = (1 + Cosθ) (1 - Cosθ)

Sinθ/(1 + Cosθ) = (1 - Cosθ)/Sinθ

By the theorem of equal ratios, we get:

(1 - Cosθ)/Sinθ = (Sinθ + 1 -Cosθ) /(1+ cosθ + Sinθ)

Squaring on both sides, we get:

[(1 - Cosθ)/Sinθ]² = (1+sinθ-cosθ / 1+sinθ+cosθ)² = LHS

= (1 - Cosθ)²/ Sin²θ  

= (1 - Cosθ)²/ (1 - cos²θ)

= (1 - Cosθ)(1 - Cosθ) / (1 - Cosθ)(1 + Cosθ)

= (1- Cosθ)/ (1 + Cosθ)

= RHS

RHS = LHS

Hence proved.