Question

How do you simplify `cot4theta-cos2theta` to trigonometric functions of a unit `theta`?

Answer 1

Use the double angle formulas.

Explanation:

The double angle formulas are given as;

`sin(2theta) = 2sin(theta) cos(theta)`
`cos(2theta) = cos^2(theta)-sin^2(theta) = 2cos^2(theta)-1 = 1-2sin^2(theta)`
`tan(2theta) = (2tan(theta))/(1-tan^2(theta))`

Notice that the double angle formulas reduce the term inside the trig function by half. If we apply the appropriate double angle formula to our function, we get;

`cot(4theta)-cos(2theta)`

`=1/tan(4theta) - (cos^2(theta)-sin^2(theta))`

`=(1-tan^2(2theta))/(2tan(2theta))-cos^2(theta)+sin^2(theta)`

Use the double angle formula again on the `tan` terms to get everything in terms of `theta`.

`(1-((2tan(theta))/(1-tan^2(theta)))^2)/(2((2tan(theta))/(1-tan^2(theta))))-cos^2(theta)+sin^2(theta)`

After some simplification, we get;

`(1-tan^2(theta))/(4tan(theta)) - tan(theta)/(1-tan^2(theta))-cos^2(theta) + sin^2(theta)`