The number of solutions of the equation 32 ^ ^ 2 x + 32 ^ ^ 2 x =...

Given, ${32}^{\tan^{2}x} + {32}^{\sec^{2}x} = 81, 0 \le x \le \dfrac{\pi}{4}$

${32}^{\tan^{2}x} + {32}^{1 + \tan^{2}x} = 81$

${32}^{\tan^{2}x} + 32 . {32}^{\tan^{2}x} = 81$

$33 \times {32}^{\tan^{2}x} = 81$

${32}^{\tan^{2}x} = \dfrac{27}{11}$

$\tan^{2}x = \ln_{32}\left(\dfrac{27}{11}\right)$
$\tan x = \sqrt{\ln_{32}\left(\dfrac{27}{11}\right)} \in \left(0, 1\right)$
One solution in \left[0, \dfrac{\pi}{4}\right]$