Question

Prove each identity? cos (x+pi/4) + cos (x-pi/4) = sqrt2 cosx

Answer 1

use the sum and difference for cos

cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

and

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

you will need to use the exact value of pi/4 for sin and cos... then simplify.

and you may also need to rationalise a denominator...

Answer 2

@campbell_st, I did as you said. However, I am a bit confused on how I would need to rationalize a denominator. Since combining like terms, I ended up with cos sqrt2/2 + cos sqrt2/2. That is the part where I am stuck on. I am not sure if I did something wrong.

Answer 3

ok so you get

\[\cos(x)\cos(\frac{\pi}{4}) - \sin(x)\sin(\frac{\pi}{4}) + \cos(x)\cos(\frac{\pi}{4}) - \sin(x) \sin(\frac{\pi}{4})\]

the terms in sin cancel each other... leaving

\[2\cos(x)\cos(\frac{\pi}{4})\]

find the exact value of cos(pi/4)

and substitute

Answer 4

oops there should be a + instead of a - before the 2nd sin(x)sin(pi/4)

Answer 5

@campbell_st, okay, I got it. I should have combined like terms first instead of distributing. Thank you!