Question

Number of real solutions of the equation $\sin a^x\cos a^x=\frac{a^x+a^{-x}}{4}$ is

• A. $1$
• B. $2$
• C. $3$
• D. $0$

Answer 1

The correct option is D $0$

Multiplying both sides with 2, we get
$2sin{a}^x.cos{a}^x=\frac{a^x+a^{-x}}{2}$
Let
$a^x=y$
Hence
$sin2y=\frac{y^2+1}{2y}$
Now
$sin2yϵ[-1,1]$
However,
$\frac{y^2+1}{2y}\ne [-1,1]$ always,
Since $\frac{y^2+1}{2y}=cosecy$
Hence range of values of RHS is not equal to LHS.
Thus no real solutions.