Total number of positive integral value of `n' such that the equations cos&

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Total number of positive integral value of `n' such that the equations

c o s^{- 1} x + (s i n^{- 1} y)^{2} = \frac{n π^{2}}{4}

and

(s i n^{- 1} y)^{2} - c o s^{- 1} x = \frac{π^{2}}{16}

are consistent, is equal to :

Options:

A . 1

B . 4

C . 3

D . 2

Answer: Option A
:
A

(s i n^{- 1} y)^{2} + c o s^{- 1} x = \frac{n π^{2}}{4}

(s i n^{- 1} y)^{2} - c o s^{- 1} x = \frac{π^{2}}{16}

\Rightarrow (s i n^{- 1} y)^{2} = \frac{(4 n + 1) π^{2}}{32}, c o s^{- 1} x = \frac{π^{2} (4 n - 1)}{32}

\Rightarrow 0 \leq \frac{(4 n + 1)}{32} π^{2} \leq \frac{π^{2}}{4}, 0 \leq \frac{(4 n - 1)}{32} π^{2} \leq π

\Rightarrow - \frac{1}{4} \leq n \leq \frac{7}{4}, \frac{1}{4}, \leq n \leq \frac{8}{π} + \frac{1}{4}

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