Question 143

if$$\ \frac{1}{sin \theta+cosec \theta}=\frac{1}{2}\ $$, then what is the value of sin$$^{100}\ \theta\ +\ cosec^{100}\ \theta$$?

Solution

Given : $$\ \frac{1}{sin \theta+cosec \theta}=\frac{1}{2}\ $$

=> $$\ \frac{1}{sin\theta+\frac{1}{sin\theta}}=\frac{1}{2}\ $$

=> $$\ \frac{sin\theta}{sin^2\theta+1}=\frac{1}{2}\ $$

=> $$sin^2\theta+1-2sin\theta=0$$

=> $$(sin\theta-1)^2=0$$

=> $$sin\theta=1$$

Also, $$cosec\theta=\frac{1}{sin\theta}=1$$

$$\therefore$$ $$\ sin^{100}\ \theta+cosec^{100}\ \theta\ $$

= $$(1)^{100}+(1)^{100}=1+1=2$$

=> Ans - (D)

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