Question

The number of solutions of
$sin^{-1} (1 + b + b^2 + ....\infty) + cos^{-1}(a - \frac{a^2}{3} + \frac{a^3}{9} ... \infty)$
$ = \frac{\pi}{2}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $\infty$

Answer 1

Answer: (D)

The given equation will be valid if
$- 1 \le 1 + b + b^2 + ... \infty \le 1$,
and $ a - \frac{a^2}{3} + \frac{a^3}{9} - ....\infty\,\in[-1, 1]$
Also, $1 + b + b^2 + .....\infty = a - \frac{a^2}{3} + \frac{a^3}{9} - ....\infty$
(Infinite G.P. series)
or $\frac{1}{1 - b} = \frac{a}{1 + \frac{a}{3}}$ or $\frac{1}{1 - b} = \frac{3a}{a + 3}$
which is true for number of values $a$ and $b$ so $\exists$ infinite solutions of the equation
$sin^{-1} (\frac{1}{1 - b}) + cos^{-1}(\frac{3a}{a + 3}) = \pi/2$