====== Question 8(x, xi & xii) Exercise 8.2 ======
Solutions of Question 8(x, xi & xii) 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(x)=====
Verify the identities: $\sec 2 x=\dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}$

** Solution. **

\begin{align*}
RHS &= \dfrac{\cos x}{\cos x+\sin x}+\dfrac{\sin x}{\cos x-\sin x}\\
&=\dfrac{\cos x(\cos x-\sin x)+\sin x(\cos x+\sin x)}{(\cos x+\sin x)(\cos x-\sin x)} \\
&= \dfrac{\cos^2 x-\sin x\cos x+\sin x\cos x+\sin^2 x}{(\cos^2 x+\sin^2 x)}\\
&= \dfrac{\cos^2 x+\sin^2 x}{\cos2 x}\\
&= \dfrac{1}{\cos2 x}\\
&= \sec 4 x\\
&=LHS
\end{align*}

=====Question 8(xi)=====
Verify the identities: $\cos ^{4} x-\sin ^{4} x=\cos 2 x$

** Solution. **

\begin{align*}
LHS &= \cos ^{4} x-\sin ^{4} x\\
&=(\cos ^{2} x+\sin ^{2} x)(\cos ^{2} x-\sin ^{2} x) \\
&=1(\cos2 x)\\
&= \cos2 x\\
&=RHS
\end{align*}

=====Question 8(xii)=====
Verify the identities: $\tan \frac{\beta}{2}+\cot \frac{\beta}{2}=2 \csc \beta$

** Solution. **

\begin{align*}
RHS &= \tan \frac{\beta}{2}+\cot \frac{\beta}{2}\\
&=\frac{\sin \frac{\beta}{2}}{\cos \frac{\beta}{2}}+ \frac{\cos \frac{\beta}{2}}{\sin \frac{\beta}{2}}\\
&=\frac{\sin^2\frac{\beta}{2}+\cos^2\frac{\beta}{2}}{\sin \frac{\beta}{2} \cos \frac{\beta}{2}}\\
&= \frac{1}{\frac{\sin \beta}{2}}\quad (by\, using\,half \, angle\, identity)\\
&=2 \csc \beta\\
&=RHS
\end{align*}


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