How to solve trigonometric sums without a calculator(C3)?

Basically, I came across this question in the past paper given below,(Question 6, part i)

http://pmt.physicsandmathstutor.com/download/Maths/A-level/C3/Papers-Edexcel/January%202013%20QP%20-%20C3%20Edexcel.pdf

How do we solve such sums? Are there special methods which we need to be aware of?

Doesn't require anything other than what you've already covered; standard trig identities, double/half angle formulae.

In this case you clearly can't do sin+cos, so multiply out and see what you can do from there. You should pick up on the fact that 22.5 is half 45.

Doesn't require anything other than what you've already covered; standard trig identities, double/half angle formulae.

In this case you clearly can't do sin+cos, so multiply out and see what you can do from there. You should pick up on the fact that 22.5 is half 45.