====== Question 8(vii, viii & ix) Exercise 8.2 ======
Solutions of Question 8(vii, viii & ix) of Exercise 8.2 of Unit 08: Fundamental of Trigonometry. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. 

=====Question 8(vii)=====
Verify the identities: $\sin 2 \theta=2 \cot \theta \sin ^{2} \theta$

** Solution. **

\begin{align*}
RHS &= 2 \cot \theta \sin ^{2} \theta\\
&= 2 \frac{\cos \theta }{\sin \theta} \sin ^{2} \theta\\
&= 2 \cos \theta \sin\theta\\
&=  \sin2 \theta\\
&=LHS
\end{align*}

=====Question 8(viii)=====
Verify the identities: $\cos ^{2} 2 x+4 \sin ^{2} x \cos ^{2} x=1$

** Solution. **

\begin{align*}
LHS &= \cos ^{2} 2 x+4 \sin ^{2} x \cos ^{2} x\\
&= \cos ^{2} 2 x+ \sin ^{2}2 x \\
&= 1\\
&=RHS
\end{align*}

=====Question 8(ix)=====
Verify the identities: $\cos 4 \theta=8 \cos ^{4} \theta-8 \cos ^{2} \theta+1$

** Solution. **

\begin{align*}
RHS &= 8 \cos ^{4} \theta-8 \cos ^{2} \theta+1\\
&=8(\frac{1+\cos2\theta}{2})^2-8(\frac{1+\cos2 \theta}{2})+1 \\
&= 2(1+\cos2\theta)^2-4(1+\cos2\theta)+1\\
&= 2(1+\cos^22\theta+2\cos2\theta)-4(1+\cos2\theta)+1\\
&= 2+2\cos^22\theta+4\cos2\theta-4-4\cos2\theta+1\\
&= 2\cos^22\theta-1\\
&=2(\frac{1+\cos 4 \theta}{2})-1\\
&=\cos 4 \theta\\
&=LHS
\end{align*}









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