Question

Which of the following statements is/are true?
Statement-I : If $cot\theta +tan\theta =2cosec\theta$, then the general value of $\theta$ is $n\pi \pm \frac{\pi }{3}$, $n\in Z$.
Statement-II : If $secxcos5x+1=0$, whrere $0<x\le \frac{\pi }{2}$, then the value of $x$ is $\frac{\pi }{3}$.

• A. Only Statement-I
• B. Only Statement-II
• C. Both Statement-I and Statement-II
• D. Neither Statement-I nor Statement-II

Answer 1

Answer: (D)

Statement-I : Given, $cot\theta +tan\theta =2cosec\theta$
$\Rightarrow \frac{cos\theta }{sin\theta }+\frac{sin\theta }{cos\theta }=\frac{2}{sin\theta }$
$\Rightarrow \frac{co{s}^2\theta +si{n}^2\theta }{sin\theta \cdot cos\theta }=\frac{2}{sin\theta }$
$\Rightarrow \frac{1}{cos\theta }=2$
$\Rightarrow cos\theta =\frac{1}{2}$
$\Rightarrow cos\theta =cos\frac{\pi }{3}$
$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3}$, where $n\in Z$

Statement-II : Given, $secxcos5x+1=0$
$\Rightarrow \frac{cos5x}{cosx}+1=0$
$\Rightarrow cos5x+cosx=0$
$\Rightarrow 2cos\left(\frac{5x+x}{2}\right)\cdot cos\left(\frac{5x-x}{2}\right)=0$
$\Rightarrow 2cosx\cdot cos2x=0$
$\Rightarrow cos3x=0$ or $cos2x=0$
$\Rightarrow cos3x=cos\frac{\pi }{2}$ or $cos2x=cos\frac{\pi }{2}$
$\Rightarrow 3x=\frac{\pi }{2}$ or $2x=\frac{\pi }{2}$
$\Rightarrow x=\frac{\pi }{6}$or $x=\frac{\pi }{4}$
Hence, the solutions are $\frac{\pi }{4}$ and $\frac{\pi }{6}$