How do you simplify cot4θ−cos2θ to trigonometric functions of a unit θ?

1 Answer
Zack M.
Dec 7, 2015

Use the double angle formulas.

Explanation:

The double angle formulas are given as;

sin(2θ)=2sin(θ)cos(θ)
cos(2θ)=cos2(θ)−sin2(θ)=2cos2(θ)−1=1−2sin2(θ)
tan(2θ)=2tan(θ)1−tan2(θ)

Notice that the double angle formulas reduce the term inside the trig function by half. If we apply the appropriate double angle formula to our function, we get;

cot(4θ)−cos(2θ)

=1tan(4θ)−(cos2(θ)−sin2(θ))

=1−tan2(2θ)2tan(2θ)−cos2(θ)+sin2(θ)

Use the double angle formula again on the tan terms to get everything in terms of θ.

1−(2tan(θ)1−tan2(θ))22(2tan(θ)1−tan2(θ))−cos2(θ)+sin2(θ)

After some simplification, we get;

1−tan2(θ)4tan(θ)−tan(θ)1−tan2(θ)−cos2(θ)+sin2(θ)

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