Question

How do you simplify `f(theta)=tan2theta-2cot2theta-3sec2theta` to trigonometric functions of a unit `theta`?

Answer 1

As explained below.

Explanation:

Write the expression as,

`sin (2theta) /cos (2 theta) - 2cos (2theta)/ sin(2 theta) -3/cos (2theta)`

`( sin^2 2theta -2 cos^2 2theta -3sin 2theta)/(sin2thetacos 2theta)`

Now replace `sin 2theta` by `2 sin theta cos theta` and

`cos 2 theta` by `cos^2 theta - sin^2 theta`

`(4sin^2 theta cos^2 theta -2(cos^4 theta -2sin^2 theta cos^2 theta +sin^4 theta) - 6sin theta cos theta)/(2 sin theta cos theta(cos^2 theta - sin^2 theta)`

`(8sin^2 theta cos^2 theta - 6 sin theta cos theta -2cos^4 theta -2sin^4 theta)/(2sin theta cos^3 theta -2cos theta sin^3 theta)`