Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • What is the value of experssion <img alt="\sin ^{2} 14^{\circ}+\sin ^{2} 23^{\circ}+\sin ^{2} 37^{\circ}+\sin ^{2} 46^{\circ}-\sin ^{2} 7^{\circ}-\sin ^{2} 16^{\circ}" src="https://learn.caree

What is the value of experssion \sin ^{2} 14^{\circ}+\sin ^{2} 23^{\circ}+\sin ^{2} 37^{\circ}+\sin ^{2} 46^{\circ}-\sin ^{2} 7^{\circ}-\sin ^{2} 16^{\circ}

Option: 1 0.75

Option: 2 0

Option: 3 0.5

Option: 4 1

Answers (1)

best_answer

Sum/Difference into Product -

Sum/Difference into Product-

 

\\1.\;\;\sin \alpha+\sin \beta=2 \sin \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)\\\\2.\;\;\sin \alpha-\sin \beta=2 \sin \left(\frac{\alpha-\beta}{2}\right) \cos \left(\frac{\alpha+\beta}{2}\right)\\\\3.\;\;\cos \alpha-\cos \beta=-2 \sin \left(\frac{\alpha+\beta}{2}\right) \sin \left(\frac{\alpha-\beta}{2}\right)\\\\4.\;\;\cos \alpha+\cos \beta=2 \cos \left(\frac{\alpha+\beta}{2}\right) \cos \left(\frac{\alpha-\beta}{2}\right)

-

\begin{array}{l}={\sin ^{2} 14^{\circ}+\sin ^{2} 23^{\circ}+\sin ^{2} 37^{\circ}+\sin ^{2} 46^{\circ}-\sin ^{2} 7^{\circ}-\sin ^{2} 16^{\circ}} \\= {\sin ^{2} 14^{\circ}+\sin ^{2} 23^{\circ}+\left(\sin ^{2} 37^{\circ}-\sin ^{2} 7^{\circ}\right)+\left(\sin ^{2} 46^{\circ}-\sin ^{2} 16^{\circ}\right)} \\ {\text { Multiplying and divide by } 2} \\ {\qquad \begin{aligned} \frac{1}{2}\left[2 \sin ^{2} 14^{\circ}+2 \sin ^{2} 23^{\circ}+2\left(\sin ^{2} 37^{\circ}-\sin ^{2} 7^{\circ}\right)+2\left(\sin ^{2} 46^{\circ}-\sin ^{2} 16^{\circ}\right)\right] \\ \Rightarrow \frac{1}{2}\left[1-\cos 28^{\circ}+1-\cos 46^{\circ}+2 \sin 44^{\circ} \sin 30^{\circ}+2 \sin 62^{\circ} \sin 30^{\circ}\right]=1 \\ \text { As } \sin 30^{\circ}=\frac{1}{2} \text { and by complementary rule } \sin 62^{\circ}=\cos 28^{\circ} ; \sin 44^{\circ}=\cos 46^{\circ} \end{aligned}}\end{array} 

Posted by

Ritika Harsh

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions