Question

Prove the following identities:

`(1 + cos theta)/(1 - cos theta) = ("cosec"  theta + cot theta)^2`

Answer 1

Given: `(1 + cos θ)/(1 - cos θ) = ("cosec"  θ + cot θ)^2`

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

Proof:

Take the Left Hand Side (LHS):

LHS = `(1 + cos θ)/(1 - cos θ)`

Rationalize the denominator:

Multiply the numerator and the denominator by (1 + cos θ):

LHS = `((1 + cos θ)(1 + cos θ))/((1 - cos θ)(1 + cos θ))`

= `(1 + cos θ)^2/(1 - cos^2θ)`

Apply the Pythagorean identity:

Use the identity 1 – cos²θ = sin²θ:

LHS = `(1 + cos θ)^2/(sin^2θ)`

= `((1 + cos θ)/(sin θ))^2`

Split the fraction:

Divide each term in the numerator by sin θ:

LHS = `(1/(sin θ) + (cos θ)/(sin θ))^2`

Substitute reciprocal and quotient identities:

Since `1/(sin θ)` = cosec θ and `(cos θ)/(sin θ) = cot θ`:

LHS = (cosec θ + cot θ)² = RHS

Thus, LHS = RHS.

Hence proved.