1 8 points Verify the following identities a cos TT COS b ta

1. (8 points) Verify the following identities a) cos TT) COS (b) tan (r tan ar-1 tan r--1 (c) cos (z y) cos (ar y) 2 cos r cos y (d) sin (8ar) 2 s cos (4r) 2. (3 points) Use the formulas for lowering powers to rewrite the expression cos (r) in terms of the first power of cosine

Solution

1) a) cos(x -pi) = -cosx

use the formula: cos(a-b) = cosacosb +sinasinb

So, cos(x -pi) = cosxcospi +sinxsinpi

= -cosx

b) tan(x -pi/4)

use the formula : tan(a -b) = [ tana - tan b]/(1 +tana*tanb)

tan(x -pi/4) = [ tanx - tanpi/4]/[ 1+ tanxtanpi/4]

= (tanx - 1)/( 1+tanx)

c) cos(x +y) +cos(x -y) = 2cosxcosy

LHS cos(x +y) +cos(x -y)

cosxcosy - sinxsiny + cosxcosy +sinxsiny

= 2cosxcosy

d) sin(8x) = 2sin4xcos4x

use the double angle identity:

sin2y = 2sinycosy

replace y = 4x

So, sin(8x) = 2sin4xcos4x = RHS