Question

The value of (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) is ______.

Answer 1

The value of (cosec θ – sin θ) (sec θ – cos θ) (tan θ + cot θ) is 1.

Explanation:

(cosec θ – sin θ) = `1/sin θ - sin θ`

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

= `1/cos θ - cos θ`

= `(1 - cos^2 θ)/cos θ` 

= `(sin^2 θ)/cos θ` 

Also `tan θ + cot θ = 1/(sin θ cos θ)`.

Multiplying: `(cos^2 θ/sin θ)·(sin^2 θ/cos θ)·(1/(sin θ cos θ))` 

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

= 1