Question
Evaluate the following:
Show that:
$\frac{\cos30^\circ+\sin60^\circ}{1+\sin30^\circ+\cos60^\circ}=\cos30^\circ$
Show that:
$\frac{\cos30^\circ+\sin60^\circ}{1+\sin30^\circ+\cos60^\circ}=\cos30^\circ$
✓
Answer
$\text{L.H.S.}=\frac{\cos30^\circ+\sin60^\circ}{1+\sin30^\circ+\cos60^\circ}=\frac{\Big(\frac{\sqrt{3}}{2}+\frac{\sqrt{3}}{2}\Big)}{\Big(1+\frac{1}{2}+\frac12\big)}$
$=\frac{\frac{\sqrt{3}+\sqrt{3}}{2}}{\frac{2+1+1}{2}}=\frac{\sqrt{3}}{2}$
Also,
$\text{R.H.S.}=\cos30^\circ=\frac{\sqrt{3}}{2}$
Hence, L.H.S. = R.H.S.
$=\frac{\frac{\sqrt{3}+\sqrt{3}}{2}}{\frac{2+1+1}{2}}=\frac{\sqrt{3}}{2}$
Also,
$\text{R.H.S.}=\cos30^\circ=\frac{\sqrt{3}}{2}$
Hence, L.H.S. = R.H.S.
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