Power-Reduction Formulas

We already know that `cos^2(t)+sin^2(t)=1` and `cos^2(t)-sin^2(t)=cos(2t)`.

Adding these formulas gives that `2cos^2(t)=1+cos(2t)` or `color(blue)(cos^2(t)=(1+cos(2t))/2)`.

Subtracting second formula from first gives that `2sin^2(t)=1-cos(2t)` or `color(green)(sin^2(t)=(1-cos(2t))/2)`.

These formulas are called power-reduction formulas. They allow us to transform `sin^2(t)` and `cos^2(t)` into expressions that contain first power of cosine of double argument.

For example, following identities hold: `sin^2(x/2)=(1-cos(x))/2`, `cos^2(pi/3+alpha)=(1+cos((2pi)/3+2alpha))/2`.

Power-reduction formulas can also be used "from left to right" for transforming sums `1+cos(2t),\ 1-cos(2t)` into product.

For example, following identities are true: `1+cos(5x)=2cos^2((5x)/2),\ 1-cos(alpha+beta)=2sin^2((alpha+beta)/2)`.

Example 1. Prove identity `tan(t/2)=(sin(t))/(1+cos(t))`.

Let's transform denominator of right side using power reduction formula and numerator of right side using double angle formula:

`(sin(t))/(1+cos(t))=(2sin(t/2)cos(t/2))/(2cos^2(t/2))=(sin(t/2))/(cos(t/2))=tan(t/2)`.

Example 2. Find `sin^4(x)+cos^4(x)` if `cos(2x)=5/13`.

We have that

`sin^4(x)+cos^4(x)=(sin^2(x))^2+(cos^2(x))^2=((1-cos(2x))/2)^2+((1+cos(2x))/2)^2=`

`=(1-2cos(2x)+cos^2(2x))/4+(1+2cos(2x)+cos^2(2x))/4=(2+2cos^2(2x))/4=(1+cos^2(2x))/2=`

`=(1+(5/13)^2)/2=(1+25/169)/2=97/169`.