Question : If $\sin heta+\operatorname{cosec} heta=2$, the value of $\sin^{n} heta+\operatorname{cosec}^{n} heta$ is:
Option 1: $2^{n}$
Option 2: $2^{\frac{1}{n}}$
Option 3: $2$
Option 4: $0$

Question

Question : If $\sin heta+\operatorname{cosec} heta=2$, the value of $\sin^{n} heta+\operatorname{cosec}^{n} heta$ is:

Option 1: $2^{n}$

Option 2: $2^{\frac{1}{n}}$

Option 3: $2$

Option 4: $0$

Team Careers360 · Accepted Answer

Correct Answer: $2$

Solution : Given: $\sin heta+\operatorname{cosec} heta=2$
Squaring both sides, we get:
$(\sin heta+cosec heta)^{2}=2^{2}$
⇒ $\sin^{2} heta+\operatorname{cosec}^{2} heta+2\sin heta×\operatorname{cosec} heta=4$
⇒ $\sin^{2} heta+\operatorname{cosec}^{2} heta=4–2$
⇒ $\sin^{2} heta+\operatorname{cosec}^{2} heta=2$
Similarly, $(\sin heta+\operatorname{cosec} heta)^{3}=2^{3}$
⇒ $\sin^{3} heta+\operatorname{cosec}^{3} heta+\sin heta×\operatorname{cosec} heta×(\sin heta+\operatorname{cosec} heta)=8$
⇒ $\sin^{3} heta+\operatorname{cosec}^{3} heta+3×2=8$
⇒ $\sin^{3} heta+\operatorname{cosec}^{3} heta=2$
Similarly, we can say the value of $\sin^{n} heta+\operatorname{cosec}^{n} heta$ will be 2.
Hence, the correct answer is $2$.

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Question : If sin⁡θ+cosecθ=2, the value of sinn⁡θ+cosecnθ is:Option 1: 2nOption 2: 21nOption 3: 2Option 4: 0

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Question : If $\sin θ + cosec θ = 2$ , the value of $\sin^{n} θ + {cosec}^{n} θ$ is:
Option 1: $2^{n}$
Option 2: $2^{\frac{1}{n}}$
Option 3: $2$
Option 4: $0$