Question

Solve the equation : $5^{\frac{1}{2}}+5^{\frac{1}{2}+{\log }_5\left(\sin x\right)}={15}^{\frac{1}{2}+{\log }_{15}\cos x}$

Answer 1

$5^{\frac{1}{2}}+5^{\frac{1}{2}+{\log }_5\left(\sin x\right)}={15}^{\frac{1}{2}+{\log }_{15}\cos x}$

$\Rightarrow 5^{\frac{1}{2}}(1+5^{{\log }_5\left(\sin x\right)})={15}^{\frac{1}{2}}.{15}^{{\log }_{15}cosx}$

$\Rightarrow 5^{\frac{1}{2}}(1+\sin x)=5^{\frac{1}{2}}.3^{\frac{1}{2}}.\cos x$

$\Rightarrow 1+\sin x=3^{\frac{1}{2}}.\cos x$

$\Rightarrow \frac{1+\sin x}{\cos x}=\sqrt{3}$

$\Rightarrow \frac{{\sin }^2\frac{x}{2}+{\cos }^2\frac{x}{2}+2\sin \frac{x}{2}\cos \frac{x}{2}}{{\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}}=\sqrt{3}$

$\Rightarrow \frac{{(\sin \frac{x}{2}+\cos \frac{x}{2})}^2}{(\cos \frac{x}{2}+\sin \frac{x}{2})(\cos \frac{x}{2}-\sin \frac{x}{2})}=\sqrt{3}$

$\Rightarrow \frac{(\sin \frac{x}{2}+\cos \frac{x}{2})}{(\cos \frac{x}{2}-\sin \frac{x}{2})}=\sqrt{3}$

Dividing both sides by $\cos \frac{x}{2}$ we get

$\Rightarrow \frac{\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}+1}{1-\frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}}=\sqrt{3}$

$\Rightarrow \frac{\tan \frac{x}{2}+\tan \frac{\pi }{4}}{1-\tan \frac{\pi }{4}\tan \frac{x}{2}}=\sqrt{3}$

$\Rightarrow \tan (\frac{x}{2}+\frac{\pi }{4})=\tan \frac{\pi }{3}$

$\Rightarrow \frac{x}{2}+\frac{\pi }{4}=n\pi +\frac{\pi }{3}$

$\Rightarrow \frac{x}{2}=n\pi +\frac{\pi }{3}-\frac{\pi }{4}$

$\Rightarrow \frac{x}{2}=n\pi +\frac{4\pi -3\pi }{12}$

$\Rightarrow \frac{x}{2}=n\pi +\frac{\pi }{12}$

$\therefore x=2n\pi +\frac{\pi }{6}$