Alg9.4.24

Find the least positive value of x such that $\log_{\cos(x)}\sin(x)+\log_{\sin(x)}cos(x)=2\,$

Let

$\log_{\cos(x)}\sin(x)=a\,$

Given problem,

$a+\frac{1}{a}=2\,$

$a^2-2a+1=0\,$

$a=1\,$

Hence

$\log_{\cos(x)}\sin(x)=1\,$

$\sin(x)=\cos(x)\,$

$\sin(x)=\sin(\frac{\pi}{2}-x)\,$

Hence

$x=\frac{\pi}{2}-x\,$

Hence $x=\frac{\pi}{4}\,$