SOLUTION: Hello,Can you please help me solve this problem: 1.Solve the following equation for x e [0, 2pi): cos^2(x+pi/2)- sin^2 (x+pi/2) =1 Thank you very much. I appreciate your

Algebra ->  Trigonometry-basics -> SOLUTION: Hello,Can you please help me solve this problem: 1.Solve the following equation for x e [0, 2pi): cos^2(x+pi/2)- sin^2 (x+pi/2) =1 Thank you very much. I appreciate your       Log On


   

Question 960613: Hello,Can you please help me solve this problem:
1.Solve the following equation for x e [0, 2pi):
cos^2(x+pi/2)- sin^2 (x+pi/2) =1
Thank you very much. I appreciate your help.
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
cos%5E2%28x%2Bpi%2F2%29-+sin%5E2%28x%2Bpi%2F2%29%22%22=%22%221

Recognize that the left side is the right side of the identity
cos%282theta%29=cos%5E2%28theta%29-sin%5E2%28theta%29 with x%2Bpi%2F2 substituted for q.

So the equation becomes:

cos%282%28x%2Bpi%2F2%29%5E%22%22%29%22%22=%22%221

cos%282x%2Bpi%29%22%22=%22%221

Adding p to an angle moves it from
1. 1st quadrant to 3rd quadrant or
2. 2nd quadrant to 4th quadrant or
3. 3rd quadrant to 1st quadrant or
4. 4th quadrant to 2nd quadrant or
5. 0 to p

In any case the cosine of an angle has the opposite sign when 
p is added to it.  IOW, cos%28theta%2Bpi%29=-cos%28theta%29 is an identity.

Using it on the left side:

-cos%282x%29%22%22=%22%221

cos%282x%29%22%22=%22%22-1

since 0%3C=x%3C2pi, multiply that by 2, 0%3C=2x%3C4pi,
so to find all solutions for x between 0 and 2p,
we must take all possible values of 2x between
0 and 4p. 

2x%22%22=%22%22matrix%281%2C3%2Cpi%2C%22%2C%22%2C3pi%29

x%22%22=%22%22matrix%281%2C3%2Cpi%2F2%2C%22%2C%22%2C3pi%2F2%29

Edwin