Question : If $\operatorname{sin} \theta=\frac{4}{5}$, find the value of $\tan \theta-\operatorname{cot} \theta$.
Option 1: $\frac{5}{12}$
Option 2: $\frac{7}{9}$
Option 3: $\frac{7}{12}$
Option 4: $\frac{7}{8}$
Correct Answer: $\frac{7}{12}$
Solution :
$\sin\theta=\frac{AB}{AC}=\frac{4}{5}$
Let $AB = 4k$ and $AC = 5k$,
Using Pythagoras theorem,
$BC = \sqrt{AC^2-AB^2}= \sqrt{(5k)^2-(4k)^2}=3k$
Now,
$\tan\theta-\cot\theta$
$= \frac{4k}{3k}-\frac{3k}{4k}$
$= \frac{7}{12}$
Hence, the correct answer is $\frac{7}{12}$.
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