Question

If `sqrt((1 - cos^2 θ)sec^2 θ) = k tan θ` and 0 < θ < 90°, then k = ______.

Answer 1

If `sqrt((1 - cos^2 θ)sec^2 θ) = k tan θ` and 0 < θ < 90°, then k = 1.

Explanation:

`sqrt((1 - cos^2 θ) sec^2 θ) = sqrt(sin^2 θ · sec^2 θ)`

= `sqrt((sin θ · sec θ)^2)`

= |sin θ·sec θ|

For 0° < θ < 90° both sin θ and sec θ are positive.

So |sin θ·sec θ| = sin θ·sec θ

= `(sin θ)/(cos θ)`

= tan θ 

Hence the left side equals tan θ, so k = 1.