====== Question 4(i-iv), Exercise 9.1 ======
Solutions of Question 4(i-iv) of Exercise 9.1 of Unit 09: Trigonometric Functions. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan. 

=====Question 4(i)=====
Check whether the function is odd or even: $y=\sin x+x \cdot \cos x$

** Solution. **

Consider $f(x)=\sin x+x \cdot \cos x$.

Take 
\begin{align*} f(-x)  = \sin (-x) + (-x)\cdot \cos (-x) \end{align*}
As we know $\sin(-x)=-\sin x$ and $\cos (-x) = \cos x$, so
\begin{align*}
f(x) & = -\sin x - x \cdot \cos x \\
& = -(\sin x + x \cdot \cos x) \\
& = -f(x)
\end{align*}
Thus the given function is odd.

=====Question 4(ii)=====
Check whether the function is odd or even: $y=x^{3} \cdot \sin x \cdot \cos x$

** Solution. **

Consider $f(x)=x^{3} \cdot \sin x \cdot \cos x$.

Take 
\begin{align*} f(-x)  = (-x)^{3} \cdot \sin (-x) \cdot \cos (-x) \end{align*}
As we know $\sin(-x)=-\sin x$ and $\cos (-x) = \cos x$, so
\begin{align*}
f(-x) & = -x^3 (-\sin x) (\cos x) \\
& = x^{3} \cdot \sin x \cdot \cos x \\
& = f(x)
\end{align*}
Thus the given function is even.

=====Question 4(iii)=====
Check whether the function is odd or even: $y=\dfrac{x^{2} \cdot \tan x}{x+\sin x}$

** Solution. **

Consider 
\[y = \frac{x^2 \cdot \tan x}{x + \sin x}.\]
Take\\ 
\[y(-x) = \frac{(-x)^2 \cdot \tan(-x)}{-x + \sin(-x)}.\]
\begin{align*}
y(-x) &= \frac{(-x)^2 \cdot \tan(-x)}{-x + \sin(-x)} \\
&= \frac{x^2 \cdot (-\tan x)}{-x - \sin x}\\
&= \frac{-x^2 \cdot \tan x}{-(x + \sin x)}\\
&= \frac{x^2 \cdot \tan x}{x + \sin x}\\
&=y(x)
\end{align*}
Thus, the given function is even.

=====Question 4(iv)=====
Check whether the function is odd or even: $y=x^{3}\sin x \cos^2 x$

** Solution. **

Consider 
\[y = x^3 \sin x \cos^2 x.\]
Take 
\[y(-x) = (-x)^3 \sin(-x) \cos^2(-x)\]
\begin{align*}
y(-x) &= (-x)^3 \cdot (-\sin x) \cdot (\cos^2 x) \\
&= -x^3 \cdot (-\sin x) \cdot \cos^2 x \\
&= -x^3 \sin x \cos^2 x\\
&=-y(x)
\end{align*}
Hence, the given function is odd.


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