Question

Prove the following:

`((1 + sin θ)^2 + ( 1 - sin θ)^2)/(2cos^2θ) = sec^2 θ + tan^2 θ`

Answer 1

Expand both squared terms in the numerator using the algebraic identities ( a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b²

(1 + sin θ)² = 1 + 2sin θ + sin² θ

(1 − sin θ)² = 1 − 2sin θ + sin² θ

= (1 + 2sin θ + sin² θ) + (1 − 2sin θ + sin² θ)

= 1 + 1 + sin² θ + sin² θ

= 2 + 2sin² θ

= `(2 + 2sin^2 θ)/(2cos^2 θ)`

= `(1 + sin^2θ)/cos^2θ`

= `1/cos^2θ + sin^2θ/cos^2θ`

`1/cos^2θ = sec^2θ`

`sin^2θ/cos^2θ = tan^2θ`

sec² θ + tan² θ