Question

Prove the following identities:

`1 + (tan^2θ)/((1 + sec θ)) = sec θ`

Answer 1

Given: `1 + (tan^2θ)/((1 + sec θ))`

To Prove: `1 + (tan^2θ)/((1 + sec θ)) = sec θ`

Proof [Step-wise]:

1. Start with the left-hand side (LHS):

LHS = `1 + (tan^2θ)/(1 + sec θ)`

2. Use the fundamental identity 1 + tan²θ = sec²θ.

Replace tan²θ by sec²θ – 1: LHS = `1 + (sec^2θ - 1)/(1 + sec θ)`.

3. Factor the numerator sec²θ – 1 as (sec θ – 1)(sec θ + 1):

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

4. Cancel the common factor (sec θ + 1) in numerator and denominator (allowed when 1 + sec θ ≠ 0):

LHS = 1 + (sec θ – 1)

5. Simplify:

LHS = 1 + sec θ – 1 

= sec θ

Thus LHS = RHS.