JEE Main 2021 (Online) 31st August Evening ShiftQuestion 1
The number of solutions of the equation $32^{\tan^{2}x}+32^{\sec^{2}x}=81$, $0\le x\le\frac{\pi}{4}$ is :
Trigonometric Ratio & Identities
Solution
Concept. With $t=\tan^2x$ and $\sec^2x=1+t$ the equation fixes $32^t$. Formulas used. (1) $\sec^2x=1+\tan^2x$Step 1 - Solve. By (1), $32^t+32\cdot32^t=33\cdot32^t=81$, so $32^t=\frac{27}{11}$ and $t=\log_{32}\frac{27}{11}\approx0.26$. Step 2 - Count. On $\left[0,\frac\pi4\right]$, $\tan^2x$ increases from $0$ to $1$, and $0.26\in[0,1]$. So there is exactly $1$ solution. Answer: (B)