Question

How do you simplify `f(theta)=sec4theta-tan4theta` to trigonometric functions of a unit `theta`?

Answer 1

`f(theta) = (1+4sin^3thetacostheta-4cos^3thetasintheta)/(sin^4theta+cos^4theta-6sin^2thetacos^2theta)`

Explanation:

Use Demoivre's and Euler's theorems to find `sin4theta` and `cos4theta`

`e^(4itheta)= cos4theta + isin4theta = (costheta + isintheta)^4 = cos^4theta + 4icos^3thetasintheta -6 cos^2thetasin^2theta -4isin^3thetacostheta + sin^4theta`

`cos4theta = "Re"(e^(4itheta)) = cos^4theta+sin^4theta-6sin^2thetacos^2theta`

`sin4theta = "Im"(e^(4itheta)) = 4cos^3thetasintheta - 4sin^3thetacostheta`

Using `sec4theta -=1/(cos4theta)` and `tan4theta-= (sin4theta)/(cos4theta)`, prove that `f(theta)` can be written in the above form.