Question

How do you simplify `f(theta)=csc2theta-sec2theta-3tan2theta` to trigonometric functions of a unit `theta`?

Answer 1

`f(theta)=(cos^2theta-sin^2theta-2costhetasintheta-4sin^2thetacos^2theta)/(2sinthetacos^3theta-sin^3thetacostheta)`

Explanation:

First, rewrite as:`f(theta)=1/sin(2theta)-1/cos(2theta)-sin(2theta)/cos(2theta)`

Then as:
`f(theta)=1/sin(2theta)-(1-sin(2theta))/cos(2theta)=(cos(2theta)-sin(2theta)-sin^2(2theta))/(sin(2theta)cos(2theta))`

We will use:
`cos(A+B)=cosAcosB-sinAsinB`
`sin(A+B)=sinAcosB+cosAsinB`

So, we get:
`f(theta)=(cos^2theta-sin^2theta-2costhetasintheta-4sin^2thetacos^2theta)/((2sinthetacostheta)(cos^2theta-sin^2theta))`

`f(theta)=(cos^2theta-sin^2theta-2costhetasintheta-4sin^2thetacos^2theta)/(2sinthetacos^3theta-sin^3thetacostheta)`