Question : If $\operatorname{cosec} heta + \operatorname{cot} heta = m$, find the value of$\frac{m^2 – 1}{m^2 + 1}$
Option 1: $1$
Option 2: $0$
Option 3: $\operatorname{cos} heta$
Option 4: $\operatorname{–sin} heta$

Question

Question : If $\operatorname{cosec} heta + \operatorname{cot} heta = m$, find the value of$\frac{m^2 – 1}{m^2 + 1}$

Option 1: $1$

Option 2: $0$

Option 3: $\operatorname{cos} heta$

Option 4: $\operatorname{–sin} heta$

Team Careers360 · Accepted Answer

Correct Answer: $\operatorname{cos} heta$

Solution : Given:
$\operatorname{cosec} heta + \cot heta = m$ ........(i)
Solution:
$\operatorname{cosec}^{2} heta-\cot^{2} heta = 1$ (Trigonometric identity)
$⇒(\operatorname{cosec} heta - \cot heta)(\operatorname{cosec} heta + \cot heta) =1$
$⇒(\operatorname{cosec} heta - \cot heta)=\frac{1}{m}$ ...........(ii)
Adding (i) and (ii), we get,
$\operatorname{cosec} heta + \cot heta + \operatorname{cosec} heta -\cot heta = m + \frac{1}{m}$
$⇒2\operatorname{cosec} heta = \frac{m^{2} + 1}{m}$
$⇒\operatorname{cosec} heta = \frac{m^{2} + 1}{2m}$
Subtracting (i) and (ii), we get,
$\operatorname{cosec} heta + \cot heta- \operatorname{cosec} heta + \cot heta = m - \frac{1}{m}$
$⇒2\cot heta = \frac{m^{2} - 1}{m}$
$⇒\cot heta = \frac{m^{2} - 1}{2m}$
$ herefore \frac{m^{2}–1}{m^{2} + 1}= \frac{m^{2} – 1}{2m}×\frac{2m}{m^2+1}=\cot heta×\frac{1}{\operatorname{cosec} heta}=\frac{\frac{\cos heta}{\sin heta}}{\frac{1}{\sin heta}}=\cos
heta$
Hence, the correct answer is $\cos heta$.

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Question : If cosecθ+cotθ=m, find the value ofm2–1m2+1Option 1: 1Option 2: 0Option 3: cosθOption 4: –sinθ

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Question : If $cosec θ + \cot θ = m$ , find the value of $\frac{m^{2} - 1}{m^{2} + 1}$
Option 1: $1$
Option 2: $0$
Option 3: $\cos θ$
Option 4: $- \sin θ$