Question

How do you express `cot^3theta-cos^2theta-tan^2theta` in terms of non-exponential trigonometric functions?

Answer 1

`(cos^5(x)-sin^5(x)-cos^4(x)sin^3(x))/(sin^3(x)*cos^2(x))`

Explanation:

We write
`1/tan^3(x)-tan^2(x)-cos^2(x)`

`(1-tan^5(x))/tan^3(x)-cos^2(x)`

`(1-tan^5(x)-tan^3(x)*cos^2(x))/tan^3(x)`

`(1-tan^5(x)-sin^2(x)/cos(x))/tan^3(x)`

`(cos(x)-cos(x)tan^5(x)-sin^3(x))/(cos(x)tan^3(x))`

`(cos^5(x)-sin^5(x)-cos^4(x)sin^3(x))/(sin^3(x)cos^2(x))`

Answer 2

`( cos 3theta + 3 sin theta }/( 3 sin theta - sin 3theta )`
`- 1/2 ( 1 + cos 2theta ) -( 1 - cos 2theta )/( 1 + cos 2theta )`

Explanation:

Use

`cos 3A = 4 cos^3A - 3 cos A and sin 3A = 3 sin A - 4 sin^3A.`

`cot^3theta - cos^2theta - tan^2theta`

`= cos^3theta/sin^3theta - cos^2theta - sin^2theta/cos^2theta`

`= ( cos 3theta + 3 sin theta }/( 3 sin theta - sin 3theta )`

`- 1/2 ( 1 + cos 2theta ) -( 1 - cos 2theta )/( 1 + cos 2theta )`