Question

Which of the following are trigonometric identities?
(Can be more then one answer)
tanx cosx cscx = 1

secx-cosx/secs=sin^2x
1-tanxtany=cos(x+y)/cosxcosy

4cosx sinx = 2cosx + 1 - 2sinx

Find all solutions to the equation cosx cos(3x) - sinx sin(3x) = 0 on the interval [0,2π]. (Points : 5)

π/8,3π/8,5π/8,7π/8,9π/8,11π/8,13π/8,15π/8

π/8,5π/8,9π/8,13π/8

3π/8,7π/8,11π/8,15π/8

π/8,9π/8, 15π/8

Answer 1

tanx cosx cscx
= sinx/cosx * cosx * 1/sinx
= 1
so the first one is an identity

For questions like this, I use this trick
pick a weird angle, e.g x = 13.79°
If that weird angle satisfies your equation, it is "highly likely" that it is an identify.
( I know that it does not prove it, but I would put a large sum of money on it)

2nd one works
3rd works
4th does not

cosx cos(3x) - sinx sin(3x0 = 0
You should recognize that pattern and get
cos(x+3x) = 0
cos (4x) = 0
4x = π/2 or 4x = 3π/2
x = π/8 or x = 3π/8
now the period of cos (4x) = 2π/4 or π/2
so by adding multiples of π/2 to any answer will give up new answers

π/8
π/8 + π/2 = 5π/8
5π/8 + π/2 = 9π/8
9π/8 + π/2 = 13π/8
etc. (numerator jumping by 4π)
same for 3π/8 , adding π/2 yields more answers

3π/8 , 7π/8, 11π/8 , 15π/8 , next one would be > 2π

looks like you answer #1, with 8 different answers.