Write each expression as at the most a single trigonometric

Write each expression as (at the most) a single trigonometric function plus possibly other expressions.

a. cos^2(3x) - sin^2(3x)
b. 2cot^2(x/3) - 2csc^2(x/3)
c. 1 - 2sin^2(2x)
d. 1 - 2sin(3theta)cos(3theta)

Solution

a.cos^2(a) - sin^2(a) =cos(2a)

cos^2(3x) - sin^2(3x) =cos(2*3x)

cos^2(3x) - sin^2(3x) =cos(6x)

b. 2cot^2(x/3) - 2csc^2(x/3) =-2(-cot^2(x/3) +csc^2(x/3))

csc^2(a) -cot^2(a)=1

2cot^2(x/3) - 2csc^2(x/3) =-2(1)

2cot^2(x/3) - 2csc^2(x/3) =-2

c. 1 - 2sin^2(a)=cos(2a)

1 - 2sin^2(2x) =cos(2*2x)

1 - 2sin^2(2x)= cos(4x)

d.2sin(a)cos(a)=sin(2a)

1 - 2sin(3theta)cos(3theta)=1-sin(2*3theta)

1 - 2sin(3theta)cos(3theta)=1-sin(6theta)