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# Prove that $\sin {{50}^{\circ }}-\sin {{70}^{\circ }}+\sin {{10}^{\circ }}=0$.

Last updated date: 16th Jul 2024
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Hint: Use the formula for $\sin A+\sin B$. Take $\sin 50+\sin 10$, simplify it using the formula and substitute it back in the equation. Use the cosine function of trigonometry to solve the rest.

We need to prove that, $\sin {{50}^{\circ }}-\sin {{70}^{\circ }}+\sin {{10}^{\circ }}=0-\left( 1 \right)$
We know the formula of $\sin A+\sin B$.
$\sin A+\sin B=2\sin \left( \dfrac{A+B}{2} \right)\cos \left( \dfrac{A-B}{2} \right)$
Let us take $\sin 50+\sin 10$, where A=50 and B=10.
$\sin 50+\sin 10=2\sin \left( \dfrac{50+10}{2} \right)\cos \left( \dfrac{50-10}{2} \right)$
\begin{align} & \sin 50+\sin 10=2\sin \left( \dfrac{60}{2} \right)\cos \left( \dfrac{40}{2} \right) \\ & \sin 50+\sin 10=2\sin 30\cos 20 \\ \end{align}
We know the value of, $\sin 30=\dfrac{1}{2}$
\begin{align} &\therefore 2\sin 30\cos 20=2\times \dfrac{1}{2}\times \cos 20=\cos 20 \\ &\therefore \sin 50+\sin 10=\cos 20-(2) \\ \end{align}
Put, $\sin 50+\sin 10=\cos 20$in equation (1).
$\therefore \cos 20-\sin 70-(4)$
By using the trigonometric cosine function,
\begin{align} & \cos \left( 90-\theta \right)=\sin \theta \\ & \cos 20=\cos \left( 90-70 \right)=\sin 70 \\ \end{align}
$\therefore$We got the value of $\cos 20=\sin 70$.
Substitute $\cos 20=\sin 70$in equation (4), we get
$\sin 70-\sin 70=0$
$\therefore$We proved that $\sin {{50}^{\circ }}-\sin {{70}^{\circ }}+\sin {{10}^{\circ }}=0$

Note: We can also solve by using the formulae,
\begin{align} & \sin \left( A+B \right)=\sin A\cos B+\cos A\sin B \\ & \sin \left( A-B \right)=\sin A\cos B-\cos A\sin B \\ \end{align}
$\sin {{50}^{\circ }}-\sin {{70}^{\circ }}+\sin {{10}^{\circ }}$can be written as,
\begin{align} & \sin \left( 60-10 \right)=\sin 60\cos 10-\cos 60\sin 10 \\ & \sin \left( 60+10 \right)=\sin 60\cos 10+\cos 60\sin 10 \\ \end{align}
$\therefore \sin \left( 60-10 \right)-\sin \left( 60+10 \right)+\sin 10=\sin 60\cos 10-\cos 60\sin 10-\sin 60\cos 10-\cos 60\sin 10+\sin 10$
[Cancel out like terms]
\begin{align} & =-2\cos 60\sin 10+\sin 10 \\ & \because \cos 60=\dfrac{1}{2} \\ & =-2\times \dfrac{1}{2}\sin 10+\sin 10=-\sin 10+\sin 10=0 \\ & \therefore \sin 50-\sin 70+\sin 10=0 \\ \end{align}