The number of solutions of tan(5π cos θ) = cot (5π sin θ) in (0,2π) is?
A) 28
B) 14
C) 4
D) 2
Hello
This equality can be written
$\displaystyle \frac{\sin(5\pi\cos\theta)}{\cos(5\pi\cos\theta)}-\frac{\cos(5\pi\sin\theta)}{\sin(5\pi\sin\theta)}= 0 $
If $\displaystyle \sin(5\pi\sin\theta)\neq 0$ and $\displaystyle \cos(5\pi\cos\theta)\neq 0$, this equation is the same as $\displaystyle \sin(5\pi\cos\theta)\sin(5\pi\sin\theta)- \cos(5\pi\sin\theta)\cos(5\pi\cos\theta)=0
$
Since $\displaystyle \cos(a+b)=\cos a \cos b-\sin a\sin b$ the equation becomes $\displaystyle \cos\left[5\pi(\sin \theta+\cos \theta) \right]=0$.
EDIT : To find the number of solutions you may need $\displaystyle \sin \theta + \cos \theta = \sqrt{2}\cos\left(\theta-\frac{\pi}{4}\right)$.