Prove the following trigonometric identities.(i) 1+cosθ+sinθ1+cosθ�

1.	Prove the following trigonometric identities.(i) 1+cosθ+sinθ1+cosθ−sinθ=1+sinθcosθ(ii) sinθ−cosθ+1sinθ+cosθ−1=1secθ−tanθ(iii) cosθ−sinθ+1cosθ+sinθ−1=cosecθ+cotθ(iv) (sinθ+cosθ)(tanθ+cotθ)=secθ+cosecθ
Answer» Prove the following trigonometric identities. (i) $\frac{1 + cos θ + sin θ}{1 + cos θ - sin θ} = \frac{1 + sin θ}{cos θ}$ (ii) $\frac{sin θ - cos θ + 1}{sin θ + cos θ - 1} = \frac{1}{sec θ - tan θ}$ (iii) $\frac{cos θ - sin θ + 1}{cos θ + sin θ - 1} = cosec θ + cot θ$ (iv) $(sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ$

Prove the following trigonometric identities.(i) 1+cosθ+sinθ1+cosθ−sinθ=1+sinθcosθ(ii) sinθ−cosθ+1sinθ+cosθ−1=1secθ−tanθ(iii) cosθ−sinθ+1cosθ+sinθ−1=cosecθ+cotθ(iv) (sinθ+cosθ)(tanθ+cotθ)=secθ+cosecθ

Answer»

Prove the following trigonometric identities.
(i) $\frac{1 + cos θ + sin θ}{1 + cos θ - sin θ} = \frac{1 + sin θ}{cos θ}$

(ii) $\frac{sin θ - cos θ + 1}{sin θ + cos θ - 1} = \frac{1}{sec θ - tan θ}$

(iii) $\frac{cos θ - sin θ + 1}{cos θ + sin θ - 1} = cosec θ + cot θ$

(iv) $(sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ$