Question

What is `sectheta*csctheta` in terms of `costheta`?

Answer 1

Step by step working is shown below.

Explanation:

`color(blue)(sec(theta)*csc(theta)`

We know

`color(green)(sec(theta) =1/cos(theta)`

`color(green)(csc(theta) = 1/sin(theta)`

`color(blue)(sec(theta)*csc(theta) = 1/cos(theta)*1/sin(theta)`

Note apply the Pythagorean identity

`color(green)(cos^2(theta)+sin^2(theta) = 1`
`color(green)(=> sin^2(theta) = (1-cos^2(theta))`

`color(green)(=>sin(theta) = sqrt(1-cos^2(theta))`

Our problem would become
`sec(theta)*csc(theta) = 1/cos(theta)*1/sin(theta)`

`color(blue)(sec(theta)*csc(theta) = 1/cos(theta)*1/sqrt(1-cos^2(theta)`

`color(blue)(sec(theta)*csc(theta) = 1/(cos(theta)sqrt(1-cos^2(theta))`