Answer
Verified
376.8k+ views
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
$\begin{align}
& {{\left( \sin 15+\cos 15 \right)}^{2}} \\
& ={{\sin }^{2}}15+{{\cos }^{2}}15+2\sin 15\cos 15 \\
& =1+\dfrac{1}{2} \\
& =\dfrac{3}{2} \\
\end{align}$
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.
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
$\begin{align}
& {{\left( \sin 15+\cos 15 \right)}^{2}} \\
& ={{\sin }^{2}}15+{{\cos }^{2}}15+2\sin 15\cos 15 \\
& =1+\dfrac{1}{2} \\
& =\dfrac{3}{2} \\
\end{align}$
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE