I was doing an advance reading on trigonometry (identities, specifically) when I encountered these problems on verification:
I'll try to edit this when I come up with answers. Thanks for those who will help!
For the last, you can start by using the following identities (derivable from the compound angle formulae):
1. sin(A + B) + sin(A - B) = 2 sin A cos B.
From which it follows that:
sin(x) + sin(3x) = 2 sin(2x) cos(x)
sin(5x) + sin(7x) = 2 sin(6x) cos(x)
2. cos(A + B) + cos(A - B) = 2 cos A cos B.
From which it follows that
cos(x) + cos(3x) = ........
cos(5x) + cos(7x) = ......
ok for problems such like this knowing the standard identities is VERY useful, a list of them can be found here
Table of Trigonometric Identities
for example lets look at the first of your problems
looking at the denominator we can write it as
so the LHS becomes
now writing tan x in terms of sine and cosine we have
then using
once again writing tan in terms of sine and cosine gives us the RHS
so all this topic is really is learning identities and practice (hence why this is deathly boring!)