Question

Prove that following identities:

$4(co{s}^3{10}^{\circ }+si{n}^3{20}^{\circ })=3(cos{10}^{\circ }+sin{20}^{\circ })$

Answer 1

Consider the L.H.S. of the given equation

$4(co{s}^3{10}^{\circ }+si{n}^3{20}^{\circ })=3(cos{10}^{\circ }+sin{20}^{\circ })$

Since $sin{30}^{\circ }=cos{60}^{\circ }=\frac{1}{2}$

and $sin{60}^{\circ }=cos{30}^{\circ }=\frac{\sqrt{3}}{2}$

$\Rightarrow sin{3.20}^{\circ }=cos{3.10}^{\circ }\Rightarrow 3sin{20}^{\circ }-4si{n}^3{20}^{\circ }=4co{s}^3{10}^{\circ }-3cos{10}^{\circ }\Rightarrow 4(co{s}^3{10}^{\circ }+si{n}^3{20}^{\circ })=3(cos{10}^{\circ }+sin{20}^{\circ })$

Hence proved.