Double and half angle formulas

The double and half angle formulas are as follows:

sin2(theta)=2sin(theta)cos(theta)

cos2(theta)=cos^2(theta)-sin^2(theta)

cos2(theta)=1-2sin^2(theta)

cos2(theta)=2cos^2(theta)-1

tan2(theta)=2tan(theta)/1-tan^2(theta)

sintheta/2=+/-square root of 1-cos(theta)/2

costheta/2=+/-square root of 1+cos(theta)/2

tantheta/2=+/-square root of 1-cos(theta)/1+cos(theta)

tantheta/2=sin(theta)/1+cos(theta)

tantheta/2=1-cos(theta)/sin(theta)

Example 1:

Simplify the given expression.

cos^2(pi/12)-sin^2(pi/12)

=cos2(pi/12)

=cos2(15)

=cos30degrees

=square root of 3/2

Example 2:

angleA is an acute angle.

If sinA=5/13, find sin2A and cos2A

x^2+5^2=13^2

x^2+25=169

x^2=144

x=12

cosA=12/13 tanA=5/12

sin2A=2(5/13)(12/13)

sin2A=120/169

cos2A=1-2(5/13)^2

=1-2(25/169)

=1-50/169

cos2A=119/169

Example3:

angleA is an acute angle.

If tanA=4/2, find cos2A and tan2A

2^2+1^2=r^2

4+1=r^2

5=r^2

r=square root of 5

sinA=square root of 5/5 cosA=2square root of 5/5

cos2A=1-2(square root of 5/5)^2

=1-2(5/25)=1-10/25

=15/25

cos2A=3/5

tan2A=2(1/2)/1-(1/2)^2=2/2=1/1-2/4

=1/1-1/2=1/1/2

tan2A=2