Question

If cosec θ + cot θ = 3, then cos θ = ______.

Answer 1

If cosec θ + cot θ = 3, then cos θ = `underlinebb(4/5)`.

Explanation:

Use the identity `("cosec"  θ - cot θ) = 1/("cosec"  θ + cot θ)`.

Given cosec θ + cot θ = 3, so cosec θ – cot θ = `1/3`.

Adding the two:

`2·"cosec"  θ = 3 + 1/3 = 10/3` 

⇒ cosec θ = `5/3`

⇒ sin θ = `3/5` 

Then `cot θ = 3 - 5/3 = 4/3`

So cos θ = cot θ · sin θ

= `(4/3) · (3/5)` 

= `4/5`