JEE Novice - (26)

Let $x = \sin\theta$ such that the top equation becomes $x + \frac{1}{x} = 2$ . From this $x$ has to be positive because otherwise if $x$ is negative then $\frac{1}{x}$ will be negative so their sum will be negative which is not the case. From this we can use the AM=GM mean. $\frac{x + \frac{1}{x}}{2} \geq \sqrt{x\frac{1}{x}} = 1 \\ x + \frac{1}{x} \geq 2$ Since equality holds when $x = \frac{1}{x}$ , for the top equation to be true, $x = 1 \implies \sin\theta = \csc\theta = 1$

1 to the power of any real number remains at 1 so $\sin^n\theta + \csc^n\theta = 1^n + 1^n \\ = \boxed{2}$