Prove that

To prove: $\sin \frac{\pi}{6} \cos 0+\sin \frac{\pi}{4} \cos \frac{\pi}{4}+\sin \frac{\pi}{3} \cos \frac{\pi}{6}=\frac{7}{4}$

Taking LHS,

$=\sin \frac{\pi}{6} \cos 0+\sin \frac{\pi}{4} \cos \frac{\pi}{4}+\sin \frac{\pi}{3} \cos \frac{\pi}{6}$

Putting $\pi=180^{\circ}$

$=\sin \frac{180}{6} \cos 0+\sin \frac{180}{4} \cos \frac{180}{4}+\sin \frac{180}{3} \cos \frac{180}{6}$

$=\sin 30^{\circ} \cos 0^{\circ}+\sin 45^{\circ} \cos 45^{\circ}+\sin 60^{\circ} \cos 30^{\circ}$

Now, we know that,

$\sin 30^{\circ}=\frac{1}{2}$

$\cos 0^{\circ}=1$

$\sin 45^{\circ}=\frac{1}{\sqrt{2}}$

$\cos 45^{\circ}=\frac{1}{\sqrt{2}}$

$\sin 60^{\circ}=\frac{\sqrt{3}}{2}$

$\cos 30^{\circ}=\frac{\sqrt{3}}{2}$

Putting the values, we get

$=\frac{1}{2} \times 1+\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}+\frac{\sqrt{3}}{2} \times \frac{\sqrt{3}}{2}$

$=\frac{1}{2}+\frac{1}{2}+\frac{3}{4}$

$=\frac{2+2+3}{4}$

$=\frac{7}{4}$

= RHS

∴ LHS = RHS

Hence Proved