Question : If $\operatorname{cos} \theta=\frac{4}{5}$, find the value of $\operatorname{cot} \theta+\tan \theta$.
Option 1: $\frac{12}{25}$
Option 2: $\frac{25}{12}$
Option 3: $\frac{27}{12}$
Option 4: $\frac{12}{27}$
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Correct Answer: $\frac{25}{12}$
Solution :
Given: $\operatorname{cos} \theta=\frac{4}{5}$
We know that, $\operatorname{sin} \theta=\sqrt{1-\operatorname{cos}^2 \theta}$
$⇒\operatorname{sin} \theta=\sqrt{1-(\frac{4}{5})^2}$
$⇒\operatorname{sin} \theta=\sqrt{1-\frac{16}{25}}$
$⇒\operatorname{sin} \theta=\sqrt{\frac{9}{25}}=\frac{3}{5}$
Now, $\operatorname{cot} \theta+\tan \theta$
$= \frac{\operatorname{cos} \theta}{\operatorname{sin} \theta}+\frac{\operatorname{sin} \theta}{\operatorname{cos} \theta}$
Putting the values, we get:
$=\frac{\frac{4}{5}}{\frac{3}{5}}+\frac{\frac{3}{5}}{\frac{4}{5}}$
$=\frac{4}{3}+\frac{3}{4}$
$=\frac{16+9}{12}$
$=\frac{25}{12}$
Hence, the correct answer is $\frac{25}{12}$.
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