If $\sin(\pi\cos\theta) = \cos(\pi\sin\theta)$,
then show that $\sin2\theta = \pm 3/4$.
I can do it simply by equating $\pi – \pi\cos\theta$ to $\pi\sin\theta$,
but that would be technically wrong as those angles could be in different quadrants. So how to solve?
[Math] If $\sin(\pi \cos\theta) = \cos(\pi\sin\theta)$, then show ……..
trigonometry
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Best Answer
$$\cos(\pi\sin\theta)=\sin(\pi\cos\theta)=\cos\left(\dfrac\pi2-\pi\cos\theta\right)$$
$$\implies\pi\sin\theta=2m\pi\pm\left(\dfrac\pi2-\pi\cos\theta\right)$$ where $m$ is any integer
$$\iff\sin\theta=2m\pm\left(\dfrac12-\cos\theta\right)$$
$$\iff\sin\theta\mp\cos\theta=2m\pm\dfrac12$$
$\sin\theta\mp\cos\theta=\sqrt2\sin\left(\theta\mp\dfrac\pi4\right)$
$\implies-\sqrt2\le\sin\theta\mp\cos\theta\le\sqrt2\implies m=?$
Now square both sides