NIMCET 2022 — Mathematics PYQ

To solve this trigonometric equation, we need to express the entire equation in terms of a single trigonometric ratio (either $\sin x$ or $\cos x$).

1. Use the Fundamental Identity:

We know that ${\cos }^{2}x=1-{\sin }^{2}x$. Let's substitute this into the given equation:

2. Simplify the Equation:

Expand the bracket:

Combine the ${\sin }^{2}x$ terms:

Subtract 4 from both sides:

3. Express as a Square of a Known Value:

We know that $\sin \left(\frac{\pi }{4}\right)=\frac{1}{\sqrt{2}}$, so ${\sin }^2\left(\frac{\pi }{4}\right)={\left(\frac{1}{\sqrt{2}}\right)}^2=\frac{1}{2}$.

Thus, the equation can be written as:

4. Find the General Solution:

The general solution for the trigonometric equation of the form ${\sin }^2\theta ={\sin }^2\alpha$ is:

Substituting our values:

where $n\in \mathbb{Z}$ (any integer).

Final Answer:

The general solution is $x=n\pi \pm \frac{\pi }{4}$.

Correct Option:

A) $x=n\pi \pm \frac{\pi }{4}$