SOLUTION: prove the identity: a) cot((x+pi)/(2)) = -tanx b) tanx = (1 - cos2x)/ (sin2x)

Algebra ->  Trigonometry-basics -> SOLUTION: prove the identity: a) cot((x+pi)/(2)) = -tanx b) tanx = (1 - cos2x)/ (sin2x)      Log On


   

Question 1155757: prove the identity:
a) cot((x+pi)/(2)) = -tanx
b) tanx = (1 - cos2x)/ (sin2x)
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
prove the identity:
a) It was supposed to be cot(x + pi/2)
cot%28x%2Bpi%2F2%29+=+-tan%28x%29
manipulate left side
cos%28+x%2Bpi%2F2%29%2Fsin%28%28x%2Bpi%29%2F2%29 .........use identities: cos%28+x%2Bpi%2F2%29=cos%28x%29%2Acos%28pi%2F2%29+-+sin%28x%29%2Asin%28pi%2F2%29, and sin%28x%2Bpi%2F2%29+=sin%28x%29+cos%28pi%2F2%29+%2B+cos%28x%29+sin%28pi%2F2%29

= ....since sin%28pi%2F2%29=1, cos%28pi%2F2%29=0

=%28cos%28x%29+%2A0-+sin%28x%29+%2A1%29%2F%28sin%28x%29+%2A0+%2B+cos%28x%29+%2A1%29+
=-+sin%28x%29%2Fcos%28x%29+
=+-tan%28x%29

so, cot%28x%2Bpi%2F2%29+=+-tan%28x%29


b)
tan%28x%29+=+%281+-+cos%282x%29%29%2F+%28sin%282x%29%29

manipulate right side:

%281+-+cos%282x%29%29%2F+%28sin%282x%29%29+
use the following identity:
cos+%282x+%29=1-2sin+%5E2+%28x+%29 and sin%282x%29=2cos+%28x%29sin%28x%29

=%281+-+%281-2sin%5E2%28x%29%29%29%2F+%282cos%28x%29sin%28x%29%29

=%281+-+1%2B2sin%5E2%28x+%29%29%2F+%282cos%28x%29sin%28x%29%29

=2sin%5E2%28x%29%2F+%282cos%28x%29sin%28x%29%29......simplify

=cross%282%29sin%5Ecross%282%29%28x%29%2F+%28cross%282%29cos%28x%29cross%28sin%28x%29%29%29

=sin%28x%29%2F+cos%28x%29

=tan%28x%29