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Find the value of the expression $2{{\left( \sin 15+\sin 75 \right)}^{2}}$?

Last updated date: 25th Jul 2024
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Hint: We first try to convert all the trigonometric ratios into forms of equal angles to apply the formulas and identities like $2\sin \theta \cos \theta =\sin 2\theta $ and ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$. We convert $\sin 75$ into $\cos 15$. We break the square part using the formula of ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$. We place the values in the formula and find the final solution.

Complete step by step answer:
We first convert all the given trigonometric ratios into forms of equal angles. We choose angles of 15.
We know that $\sin \alpha =\cos \left( \dfrac{\pi }{2}-\alpha \right)$. Putting the value of $\alpha =75$, we get
$\sin 75=\cos \left( \dfrac{\pi }{2}-75 \right)=\cos 15$.
Therefore, we have $2{{\left( \sin 15+\sin 75 \right)}^{2}}=2{{\left( \sin 15+\cos 15 \right)}^{2}}$.
We now apply the formula of ${{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab$.
We get ${{\left( \sin 15+\cos 15 \right)}^{2}}={{\sin }^{2}}15+{{\cos }^{2}}15+2\sin 15\cos 15$.
We have the formula of multiple angles where we get $2\sin \theta \cos \theta =\sin 2\theta $.
So, we get $2\sin 15\cos 15=\sin \left( 15\times 2 \right)=\sin 30=\dfrac{1}{2}$.
We also have the identity formula of ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta =1$.
Applying the formula, we get ${{\sin }^{2}}15+{{\cos }^{2}}15=1$.
Putting all the values we get
  & {{\left( \sin 15+\cos 15 \right)}^{2}} \\
 & ={{\sin }^{2}}15+{{\cos }^{2}}15+2\sin 15\cos 15 \\
 & =1+\dfrac{1}{2} \\
 & =\dfrac{3}{2} \\
At the end we multiply with 2 to get $2{{\left( \sin 15+\sin 75 \right)}^{2}}=2\times \dfrac{3}{2}=3$.
The value of the expression $2{{\left( \sin 15+\sin 75 \right)}^{2}}$ is 3.

Note: we can also convert $\sin 15$ into $\cos 75$. But in that case the multiple angle formula gives us the $\sin \left( 75\times 2 \right)=\sin 150$ instead of $\sin 30=\dfrac{1}{2}$. We have to convert the associative angle using other formulas to simplify it. The problem becomes unnecessarily longer and that’s why we used an angle of 15.