Question

If $x=30{}^\circ$ verify that $\sin x=\sqrt{\dfrac{1-\cos 2x}{2}}$.

Answer 1

Hint : The left-hand side and right-hand side are calculated by putting the value of x. The right-hand side can be obtained by using the value of x.
Formulas used:
The value of $\sin 30{}^\circ =\dfrac{1}{2}$ and the value of $\cos 60{}^\circ =\dfrac{1}{2}$.

Complete step-by-step answer :
First step will be considering $\sin x$ as the left-hand side and put the value of x.
The right-hand side be $\sqrt{\dfrac{1-\cos 2x}{2}}$ and put the value of x, and obtain the result.
The verification can be obtained as,

& \sin x=\sqrt{\dfrac{1-\cos 2x}{2}} \\\ & L.HS.=\sin x \\\ & L.HS.=\sin 30{}^\circ =\dfrac{1}{2} \\\ & R.H.S=\sqrt{\dfrac{1-\cos 2x}{2}} \\\ & R.H.S=\sqrt{\dfrac{1-\cos \left( 2\cdot 30 \right){}^\circ }{2}} \\\ & R.H.S=\sqrt{\dfrac{1-\cos 60{}^\circ }{2}} \\\ & R.H.S=\sqrt{\dfrac{1-\dfrac{1}{2}}{2}} \\\ & R.H.S=\sqrt{\dfrac{\dfrac{1}{2}}{2}} \\\ & R.H.S=\sqrt{\dfrac{1}{4}} \\\ & R.H.S=\dfrac{1}{2} \\\ \end{aligned}$$ $$Therefore,\text{ }L.H.S=R.H.S\text{ }\left[ verified \right]$$. Thus, the above expression is verified. **Note** : The standard identities of trigonometric ratio have some special value for standard angle. The value of the trigonometric ratios is different for different angles.