Question

Statement I: lf $f\left(\theta \right)=\cos {\theta }_1.\cos {\theta }_2\dots .\cos {\theta }_n$, then the value of $\tan {\theta }_1+\tan {\theta }_2+\dots +\tan {\theta }_n=-\frac{f^{\prime }\left(\theta \right)}{f\left(\theta \right)}$
Statement II: Differential coefficient of the function $f\left(g\right(x\left)\right)$ w.r.t function $g\left(x\right)$ is $f^{\prime }\left(g\right(x\left)\right)$.

Which of the above statements is/are correct?

• A. Statement $I$ only
• B. Statement $II$ only
• C. Both $I$ and $II$
• D. Neither $I$ nor $II$

Answer 1

Answer: (C)

Both $I$ and $II$

I: $f\left(\theta \right)=\cos {\theta }_1\cdot \cos {\theta }_2\cdots \cos {\theta }_n$

Differentiating both sides,

$f^{\prime }\left(\theta \right)=-\sin {\theta }_1\cos {\theta }_2\cdots \cos {\theta }_n-\cos {\theta }_1\sin {\theta }_2\cdots \cos {\theta }_n\cdots \cdots -\cos {\theta }_1\cos {\theta }_2\cdots sin{\theta }_n$

Dividing $f^{\prime }\left(\theta \right)$ by $f\left(\theta \right)$, we get
$\frac{f^{\prime }\left(\theta \right)}{f\left(\theta \right)}=-\tan {\theta }_1-\tan {\theta }_2-\tan {\theta }_3-\cdots -\tan {\theta }_n$
$-\frac{f^{\prime }\left(\theta \right)}{f\left(\theta \right)}=\tan {\theta }_1+\tan {\theta }_2+\tan {\theta }_3-\cdots \cdots -\tan {\theta }_n$

II: $\frac{d\left(f\right(g\left(x\right)\left)\right)}{d\left(g\right(x\left)\right)}=\frac{d\left(f\right(g\left(x\right)\left)\right)}{d\left(x\right)}\times \frac{d\left(x\right)}{d\left(g\right(x\left)\right)}=\frac{f^{\prime }\left(g\right(x\left)\right)g^{\prime }\left(x\right)}{g^{\prime }\left(x\right)}=f^{\prime }\left(g\right(x\left)\right)$

Both I and II are correct.