Question

1. Find the exact value by using a half-angle identity. (1 point)

Answer 1

1. Find the exact value by using a half-angle identity. 
sin 22.5°

Using the half angle formula you get:
sin2(θ)=12[1−cos(2θ)]
if θ=22.5° then 2θ=45°
so you get:
sin2(22.5°)=12[1−cos(45°)]
sin2(22.5°)=12[1−√2/2]=2−√24
and square root both sides:
sin(22.5°)=±√2−√24=±0.382
so 

sin(22.5°)=0.382the answer is the letter D) one half times the square root of quantity two minus square root of two

2. Verify the identity.

cot x minus pi divided by two. = -tan x

Cot(x-pi/2)=-tan(x)

 sin(A − B) = sin A cos B − cos A sin B

sin(x – pi/2) = sin x cos (pi/2) − cos x sin (pi/2)=-cosx

 

cos(A − B) = cos A cos B − sin A sin B

cos(x− pi/2) = cos x cos pi/2 − sin x sin pi/2=-sinx

 Cot(x-pi/2)=cos(x-pi/2)/sin(x-pi/2)

= (-sinx)/(-cosx)=-tanx--------------ok