Question

Prove :

(tan θ + sec θ - 1)/(tan θ - sec θ + 1) = (1 + sin θ)/cos θ

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Answer 1

Solution :

Let A = (tan θ + sec θ - 1)/(tan θ - sec θ + 1) and

B = (1 + sin θ)/cos θ.

A = (tan θ + sec θ - 1)/(tan θ - sec θ + 1)

A = [(tan θ + sec θ) - (sec2θ - tan2θ)]/(tan θ - sec θ + 1)

A = {(tan θ + sec θ) (1 - sec θ + tan θ)}/(tan θ - sec θ + 1)

A = {(tan θ + sec θ) (tan θ - sec θ + 1)}/(tan θ - sec θ + 1)

A = tan θ + sec θ

A = (sin θ/cos θ) + (1/cos θ)

A = (sin θ + 1)/cos θ

A = (1 + sin θ)/cos θ

A = B, (Proved)