Double-Angle & Half-Angle Formulas!

**These would be your Double and Half Angle Formulas:
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.
2cos^210degrees-1
=cos2(10)=cos20degrees

Example 2:
Simplify the given expression.
4tanB/1-tan^2B
=2(2tanB/1-tan^2B)
=tan2(B)=2tan2B

Example 3:
Simplify the given expression.
2sin35degrees(cos35degrees)
=sin2(35degrees)
=sin70degrees

Example 4:
Simplify the given expression.
2tan25dgrees/1-tan^2(25degrees)
=tan2(25)
=tan50degrees

Example 5:
Simplify the given expression.
1-2sin^2(x/2)
=cos2(x/2)
=cosx

Example 6:
Simplify the given expression.
square root of 1-cos80degrees/2
=sin(theta/2)
=sin(80/2)
=sin40degrees

Example 7:
Simplify the given expression.
2cos(2pi/8)-1
=cos2(pi/8)
=cos2(45/2)
=cos45
=square root of 2/2

Example 8:
Simplify the given expression.
cos^2(pi/12)-sin^2(pi/12)
=cos2(pi/12)
=cos2(15)
=cos30degrees
=square root of 3/2

Example 9:
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

Example 10:
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