Question

Prove the following identities:

sin²θ tan θ + cos²θ cot θ + 2 sin θ cos θ = tan θ + cot θ

Answer 1

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

= `((sin^3θ)/(cos θ) + (cos^3θ)/(sin θ) + 2 sin θ cos θ)`

= `(sin^4θ + cos^4θ + 2sin^2θ cos^2θ)/(cos θ sin θ)`

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

= `1/(cos θ sin θ)`

= (sec θ cosec θ)

RHS = `((sin θ)/(cos θ) + (cos θ)/(sin θ))`

= `((sin^2θ + cos^2θ))/(cos θ sin θ)`

= `1/(cos θ sin θ)`

= sec θ cosec θ

∴ LHS = RHS.