Question

How do you simplify `f(theta)=csc2theta-sin2theta-3cot2theta` to trigonometric functions of a unit `theta`?

Answer 1

`f(theta)=1/(2sinthetacostheta)-2sinthetacostheta-(3(1-tan^2theta))/(2tantheta)`

Explanation:

Know the following identities:

• `sin2theta=2sinthetacostheta`
• `tan2theta=(2tantheta)/(1-tan^2theta)`

From here, we can determine the cosecant and cotangent double angle formulas, since `csctheta=1/sintheta` and `cottheta=1/tantheta`.

• `csc2theta=1/(2sinthetacostheta)`
• `cot2theta=(1-tan^2theta)/(2tantheta)`

Thus,

`f(theta)=1/(2sinthetacostheta)-2sinthetacostheta-(3(1-tan^2theta))/(2tantheta)`