SOLUTION: Prove the following trig identities: a) tan 2x- 2tan 2x sin^2x= sin 2 x b) sec^2x-2sec x cos x + cos^2x= tan^2x- sin^2 x

Algebra ->  Trigonometry-basics -> SOLUTION: Prove the following trig identities: a) tan 2x- 2tan 2x sin^2x= sin 2 x b) sec^2x-2sec x cos x + cos^2x= tan^2x- sin^2 x       Log On


   

Question 1183668: Prove the following trig identities:
a) tan 2x- 2tan 2x sin^2x= sin 2 x
b) sec^2x-2sec x cos x + cos^2x= tan^2x- sin^2 x
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

Prove the following trig identities:
a)
tan+%282x%29-+2tan+%282x%29%2A+sin%5E2%28x%29=+sin%28+2x%29

manipulate left side

tan+%282x%29-+2tan+%282x%29%2A+sin%5E2%28x%29........factor

=tan+%282x%29%281-+2sin%5E2%28x%29%29...........use+tan%282x%29=sin%282x%29%2Fcos%282x%29+and+1+-+2sin%5E2%28x%29+=+cos%282x%29

=+%28sin%282x%29%2Fcos%282x%29%29%28cos%282x%29%29

= %28sin%282x%29+%2Acos%282+x%29%29%2Fcos%282+x%29.........simplify

= sin%282x%29


b)

sec%5E2%28x%29-2sec+%28x%29+cos%28+x%29+%2B+cos%5E2%28x%29=+tan%5E2%28x%29-+sin%5E2+%28x%29

manipulate left side

sec%5E2%28x%29-2sec%28x%29%2A+cos%28+x%29+%2B+cos%5E2%28x%29..........sec%28x%29=1%2Fcos%28+x%29

=%281%2Fcos%5E2%28+x%29%29-2%281%2Fcos%28x%29%29%2A+cos%28x%29+%2B+cos%5E2%28x%29............cos%5E2%28x%29=1-sin%5E2+%28x%29
=1%2Fcos%5E2%28+x%29-2+%2B1-sin%5E2%28x%29

=1%2Fcos%5E2%28+x%29-1-sin%5E2%28x%29

=%281%2Fcos%5E2%28+x%29-1%29-sin%5E2%28x%29

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

=%28%281-cos%5E2%28x%29%29%2Fcos%5E2%28x%29%29-sin%5E2%28x%29............1-cos%5E2%28x%29=sin%5E2%28x%29

=sin%5E2%28x%29%2Fcos%5E2%28x%29-sin%5E2%28x%29

=tan%5E2%28x%29-sin%5E2%28x%29