Answer:
sqrt(2)/2
Step-by-step explanation:
Given tan(x)=2-cot(x), find sin(x).
Rewrite in terms of sine and cosine:
sin(x)/cos(x)=2-cos(x)/sin(x)
Multiply both sides by cos(x)sin(x):
sin^2(x)=2sin(x)cos(x)-cos^2(x)
Rewrite cos^2(x) using the identity sin^2(x)+cos^2(x)=1:
sin^2(x)=2sin(x)cos(x)-(1-sin^2(x))
Distribute:
sin^2(x)=2sin(x)cos(x)-1+sin^2(x)
Subtracting sin^2(x) on both sides:
0=2sin(x)cos(x)-1
Add 1 on both sides:
1=2sin(x)cos(x)
Use identity sin(2x)=2sin(x)cos(x) to rewrite right:
1=sin(2x)
Since sin(pi/2)=1, then 2x=pi/2.
Dividing both sides by 2 gives x=pi/4.
So sin(pi/4)=sqrt(2)/2
