Question

Sintheta/1+costheta+1+costheta/sintheta=2 cosec theta

Answer 1

Trigonometry:

Trigonometry is the study of relations between angles and their ratios with various properties to find sine, cosine, tan, cosec, sec and cot ratios. Some properties can deduce angles in their general value apart from known definite ones. Now let us know some identities-

1. sin²θ + cos²θ = 1

2. sec²θ - tan²θ = 1

3. cosec²θ - cot²θ = 1

4. sinθ . cosecθ = 1

5. cosθ . secθ = 1

6. tanθ . cotθ = 1

Proof:

L.H.S. = sinθ/(1 + cosθ) + (1 + cosθ)/sinθ

= {sin²θ + (1 + cosθ)²} / {sinθ (1 + cosθ)}

= (sin²θ + 1 + 2 cosθ + cos²θ) / {sinθ (1 + cosθ)}

= (2 + 2 cosθ) / {sinθ (1 + cosθ)} ,

since sin²θ + cos²θ = 1

= 2 (1 + cosθ) / {sinθ (1 + cosθ)}

= 2 / sinθ

= 2 cosecθ = R.H.S.

Hence, proved.