====== Question 8(i, ii & iii) Exercise 8.2 ======
Solutions of Question 8(i, ii & iii) 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(i)=====
Verify the identities: $(\sin \theta+\cos \theta)^{2}=1+\sin 2 \theta$

** Solution. **
\begin{align*}
LHS & = (\sin \theta+\cos \theta)^{2} \\
&=\sin^2\theta + \cos^2\theta +2\sin \theta \cos\theta\\
&= 1+2\sin \theta \cos\theta \quad (\because \sin^2\theta+\cos^2\theta=1) \\
&=1+\sin 2\theta \quad (\because \sin2\theta=2\sin\theta \cos\theta) \\
&=RHS
\end{align*}
GOOD




=====Question 8(ii)=====
Verify the identities: $\tan 2 x=\dfrac{1}{1-\tan x}-\dfrac{1}{1+\tan x}$

** Solution. **

\begin{align*}
RHS & = \frac{1}{1-\tan x}-\frac{1}{1+\tan x} \\
&= \frac{1+\tan x-1+\tan x}{(1-\tan x)(1+\tan x)} \\
&= \frac{2\tan x}{1-\tan^2 x} \\
&=\tan 2x \\
& = LHS
\end{align*}







=====Question 8(iii)=====
Verify the identities: $\tan \frac{\theta}{2}=\frac{\sin \theta}{1+\cos \theta}$

** Solution. **

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


\end{align*}














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