Question

If $7\text{cosec}\theta -3\cot \theta =7$, find the value of $7\cot \theta -3\text{cosec}\theta$

• A. $5$
• B. $3$
• C. $\frac{7}{3}$
• D. $\frac{3}{7}$

Answer 1

Answer: (B)

$3$

Given, $7\text{cosec}\theta -3\cot \theta =7$

$(7\text{cosec}\theta -3\cot \theta {)}^2=49$

On simplifying, we get

$49{\text{cosec}}^2\theta -42\cot \theta .\text{cosec}\theta +9{\cot }^2\theta =49$

$\Rightarrow 49(1+{\cot }^2\theta )-42\text{cosec}\theta .\cot \theta +9{\cot }^2\theta =49$

$\Rightarrow 42\text{cosec}\theta .\cot \theta =58{\cot }^2\theta$ ....(1)

Now, let $7\cot \theta -3\text{cosec}\theta =x$

Thus $x^2=49{\cot }^2\theta +9{\text{cosec}}^2\theta -42\text{cosec}\theta .\cot \theta$

$\Rightarrow x^2=58{\cot }^2\theta +9-58{\cot }^2\theta$ (using (1))

$\Rightarrow x^2=9$

$\Rightarrow x=3$