Question

Sum of all the solutions of the given equations ${\tan }^{2}33x=\cos 2x-1$ which lie in the interval $[0,314]$ is

• A. $5050\pi$
• B. $4950\pi$
• C. $5151\pi$
• D. None of these

Answer 1

The correct option is B $4950\pi$

$ta{n}^{2}33x=cos2x-1$

As we know that $cos2x-1=-2si{n}^{2}x$
Therefore, $ta{n}^{2}33x=-2si{n}^{2}x$

LHS is always positive, RHS is always negative
Solution or equality exists only when both sides are 0

Therefore $x=n\pi$

Now $314<100\pi$

Solutions = $0,\pi ,2\pi ,....99\pi$

Sum = $\pi (1+2+\mathrm{3...99})=\frac{99(99+1)}{2}\pi$

answer is $4950\pi$