SOLUTION: Prove each of the following trigonometric identities. 1) sin x sin 2x + cox x cos 2x = cos x 2) cot x = sin x sin (pi/2 - x) + cos^2 x cot x 3) 2 csc 2x = sec x csc x

Algebra ->  Trigonometry-basics -> SOLUTION: Prove each of the following trigonometric identities. 1) sin x sin 2x + cox x cos 2x = cos x 2) cot x = sin x sin (pi/2 - x) + cos^2 x cot x 3) 2 csc 2x = sec x csc x       Log On


   



Question 1183638: Prove each of the following trigonometric identities.
1) sin x sin 2x + cox x cos 2x = cos x
2) cot x = sin x sin (pi/2 - x) + cos^2 x cot x
3) 2 csc 2x = sec x csc x

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Prove each of the following trigonometric identities.
1)

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

manipulate left side

sin+%28x%29+%2Asin+%282x%29+%2B+cos+%28x+%29%2Acos+%282x%29 ..........use sin+%282x%29+=2sin%28x%29%2A+cos%28x%29+and+cos+%282x%29=cos%5E2%28x%29+-+sin%5E2%28x%29

=

= 2sin%5E2%28x%29%2A+cos%28x%29%2Bcos+%28x+%29%2A+%28cos%5E2%28x%29+-+sin%5E2%28x%29%29...factor out cos%28x%29

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

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

= cos%28x%29%28sin%5E2%28x%29%2B+cos%5E2%28x%29+%29............use sin%5E2%28x%29%2B+cos%5E2%28x%29=1

= cos%28x%29


2)

cot+%28x%29+=+sin+%28x+%29sin+%28pi%2F2+-+x%29+%2B+cos%5E2%28+x%29+cot%28+x%29

manipulate right side

sin+%28x+%29sin+%28pi%2F2+-+x%29+%2B+cos%5E2%28+x%29+cot%28+x%29.........use

=sin+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+cot%28+x%29.... use cot%28+x%29=cos%28x%29%2Fsin+%28x+%29

=sin+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+%28cos%28x%29%2Fsin+%28x+%29%29

=%28sin%5E2+%28x+%29cos%28x%29+%2B+cos%5E2%28+x%29+%2Acos%28x%29%29%2Fsin+%28x+%29%29

=%28%28sin%5E2+%28x+%29%2B+cos%5E2%28+x%29%29+%2Acos%28x%29%29%2Fsin+%28x+%29%29

=%281+%2Acos%28x%29%29%2Fsin+%28x+%29%29

=cos%28x%29%2Fsin+%28x+%29

=cot%28+x%29


3)

2csc+%282x%29+=+sec+%28x%29+%2Acsc%28+x%29

manipulate left side

2csc+%282x%29 ..................use csc+%282x%29+=+%281%2F2+%29csc%28x%29+sec%28x%29

=2%281%2F2%29csc%28x%29+%2Asec%28x%29%29 .....simplify

=csc%28x%29+%2Asec%28x%29%29