Question

Prove the following:

`(1 + tan θ)/sin θ + (1 + cot θ)/cos θ` = 2(sec θ + cosec θ)

Answer 1

tan θ = `sin θ/cos θ and cot θ = cos θ/sin θ`

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

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

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

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

= `(2(sin θ + cos θ))/(sin θ.cos θ)`

LHS = `(2 sin θ)/(sin θ.cos θ) + (2 cos θ)/(sin θ.cos θ)`

LHS = `2/cos θ + 2/sin θ`

sec θ = `1/cos θ` and cosec θ = `1/sin θ`

LHS = 2 sec LHS = + 2 cosec θ

LHS = RHS