Question

Prove that: `(1 + "cosec"  θ)/("cosec"  θ) = (cos^2 θ)/(1 - sin θ)`

Prove the following trigonometric identities:

`(1 + "cosec"  θ)/("cosec"  θ) = (cos^2 θ)/(1 - sin θ)`

Answer 1

Simplify the LHS:

cosecθ in terms of sinθ, where cosecθ = `1/sinθ`

`LHS = (1+1/sintheta)/(1/sintheta)`

`LHS = ((sintheta+1)/sintheta)/(1/sintheta)`

Cancel out sinθ from the numerator and the denominator:

LHS = sinθ + 1

Simplify the RHS:

`RHS = cos^2theta/(1-sintheta)`

`RHS = (1-sin^2theta)/(1-sintheta)`

`RHS = ((1-sintheta)(1+sintheta))/(1-sintheta)`

RHS = 1 + sinθ

Since the simplified form of both the LHS and the RHS is is 1 + sinθ, we have proven that LHS = RHS