# [SPM Add Math Trigonometry question] How to prove question (ii)

### Question

(ii) Buktikan identiti \sin \left(\alpha-30^{\circ}\right)+\operatorname{kos}\left(\alpha+30^{\circ}\right)=\frac{\sqrt{3}-1}{\sqrt{2}}\sin \left(\theta+45^{\circ}\right)
Prove the identity \sin \left(\alpha-30^{\circ}\right)+\cos \left(\alpha+30^{\circ}\right)=\frac{\sqrt{3}-1}{\sqrt{2}} \sin \left(\theta+45^{\circ}\right)

### Solution

Hello Farhaini, your first step is correct. I think you got a bit lost after expanding using the double angle formula, notice that you can factorize after expanding.

And here’s the trick: split the 2 in the denominator to \sqrt{2} \times\sqrt{2} ! Why? Because at the RHS of the equation, the expression is sin(\theta + 45) , and the denominator is a \sqrt 2.

Also, I’m not sure why did the question use \theta instead of \alpha on the RHS, the unknown shouldn’t change though, could be typing error.

Hope this helps!

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Ohhh Thank you very much !

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