Question

The number of solutions of the equation ${32}^{{\tan }^{2}x}+{32}^{{\sec }^{2}x}=81,$ $0\le x\le \frac{\pi }{4}$ is

• A. $3$
• B. $1$
• C. $2$
• D. $0$

Answer 1

The correct option is B $1$

${32}^{{\tan }^{2}x}+{32}^{1+{\tan }^{2}x}=81$
$\Rightarrow {32}^{{\tan }^{2}x}\left(33\right)=81$
$\Rightarrow {32}^{{\tan }^{2}x}=\frac{81}{33}$
If $x\in [0,\frac{\pi }{4}]$ then ${32}^{{\tan }^{2}x}\in [1,32]$ and is always increasing
Hence one solution only.