Solve this
Question:

If $\tan \theta=\frac{8}{15}$ then $\operatorname{cosec} \theta=?$

(a) $\frac{17}{8}$

(b) $\frac{8}{17}$

(c) $\frac{15}{17}$

(d) $\frac{17}{15}$

Solution:

$\tan \theta=\frac{8}{15}$

$\Rightarrow \frac{1}{\tan \theta}=\frac{15}{8}$

$\Rightarrow \cot \theta=\frac{15}{8} \quad \ldots \ldots(1)$

Now,

$\operatorname{cosec}^{2} \theta=1+\cot ^{2} \theta$

$\Rightarrow \operatorname{cosec}^{2} \theta=1+\left(\frac{15}{8}\right)^{2} \quad$ [Using (1)]

$\Rightarrow \operatorname{cosec}^{2} \theta=1+\frac{225}{64}=\frac{289}{64}$

$\Rightarrow \operatorname{cosec} \theta=\sqrt{\frac{289}{64}}=\frac{17}{8}$

Hence, the correct answer is option (a).

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