Help in Solving a Trigonometric Equation

Question

Solve the equation $$\left(\sin x + \cos x\right)^{1+\sin(2x)} = 2$$ when $-\pi \le x \le \pi $ .

I have tried to use $\sin (2x) = 2\sin x \cos x$ identity but I this doesn't lead me to a conclusion.

I will appreciate the help.

Answer 1

Hint: We have $\sin x+\cos x=\sqrt{2}\sin(x+\pi/4)$. It is not easy for a small positive power of this to be $2$.

Answer 2

Setting $\displaystyle x+\frac\pi4=y, \sin2x=\sin2\left(y-\frac\pi4\right)=-\sin\left(\frac\pi2-2y\right)=-\cos2y$

$\displaystyle(\cos x+\sin x)^{1+\sin2x}=(\sqrt2\sin y)^{2\sin^2y}=\left((\sqrt2\sin y)^2\right)^{\sin^2y}=(2\sin^2y)^{\sin^2y}$

Now, $\displaystyle0\le\sin^2y\le1$

So, $\displaystyle(2\sin^2y)^{\sin^2y}$ will be $=2$ if $\displaystyle\sin^2y=1\iff\cos y=0\iff y=(2n+1)\frac\pi2$ where $n$ is any integer

Answer 3

To solve $(\sin x+\cos x)^{1+\sin 2x}=2$ when $−\pi\leq x\leq \pi$. Let $z=\sin x+\cos x$, then $z^2=1+\sin 2x$. Hence the equation is equivalent to $z^{z^2}=2$ which has the only possible solution $z=\sqrt{2}$.

Hence $\sin x+\cos x=\sqrt{2}\sin(x+\frac{\pi}{4})=\sqrt{2}$.

Thus $\sin(x+\frac{\pi}{4})=1$, which implies $x+\frac{\pi}{4}=\frac{\pi}{2}$. Hence $x=\frac{\pi}4$.

Or using $z^2=1+\sin 2x$ with $z=\sqrt{2}$ gives $1+\sin 2x=2$

Hence $\sin 2x=1$ implies $2x=\frac{\pi}{2}$. Hence $x=\frac{\pi}4$.