Question

How do you simplify `cos8theta-5tan2theta` to trigonometric functions of a unit `theta`?

Answer 1

`cos 8theta-5 tan 2theta=`
`{2[2(2 cos^2 theta-1)^2-1]^2-1}-(10tan theta)/(1-tan^2 theta)`

Explanation:

Use double angle formulas for tangent

`tan 2A= (2 tan A)/(1-tan^2 A)`

Use half-angle formulas for cosine

`cos (A/2)=sqrt((1+cos A)/2`

so that

`cos A=2*cos^2 (A/2)-1`

We can now write

`cos 8theta=2 cos^2 4theta-1`

`cos 4theta=2 cos^2 2theta-1`

`cos 2theta=2 cos^2 theta-1`

It follows that

`cos 8theta-5 tan 2theta=`
`{2[2(2 cos^2 theta-1)^2-1]^2-1}-(10tan theta)/(1-tan^2 theta)`

God bless...I hope the explanation is useful.