A-Level Maths Trig Identities Question

Q1) Use the compound angle formulae to write sin165 degrees as surds
I did sin(A+B) = sinAcosB =sinAcosB + cosAsinB
sin(45+120) = sin45cos120 + cos45sin120
= square root 6 - square root 2 / 4
but the answer says square root 3 - 1 / 2 square root 2

$\dfrac{\sqrt 6-\sqrt 2}{4}=\dfrac{\sqrt 3 - 1}{2\sqrt 2}$

from dividing top and bottom by $\sqrt 2$

$\dfrac{\sqrt 6-\sqrt 2}{4}=\dfrac{\sqrt 3 - 1}{2\sqrt 2}$

from dividing top and bottom by $\sqrt 2$

wait why do we divide by root 2

To get to the answer they give.

The answer you had was perfectly correct - unless they requested a specific form.

oh, but what if we didn't know the answer ..

If they haven't specified a particular form, they can't mark you down for presenting it in a format that differs to theirs.

Your real issue here was in not recognising that your answer and theirs were the same.