NIMCET 2024 — Mathematics PYQ

We need to evaluate the expression:

$$E=\tan \left(\frac{\pi }{4}+\theta \right)\tan \left(\frac{3\pi }{4}+\theta \right)$$

Step 1: Simplify the second term using Allied Angles

We can rewrite the angle $\frac{3\pi }{4}+\theta$ in terms of $\pi$:

$$\frac{3\pi }{4}+\theta =\pi -\left(\frac{\pi }{4}-\theta \right)$$

Using the trigonometric identity $\tan (\pi -A)=-\tan (A)$:

$$\tan \left(\pi -\left(\frac{\pi }{4}-\theta \right)\right)=-\tan \left(\frac{\pi }{4}-\theta \right)$$

Substitute this back into the expression:

$$E=-\tan \left(\frac{\pi }{4}+\theta \right)\tan \left(\frac{\pi }{4}-\theta \right)$$

Step 2: Apply the compound angle formula for tangent

Recall the expansion formulas:

$$\tan (A+B)=\frac{\tan A+\tan B}{1-\tan A\tan B}$$

$$\tan (A-B)=\frac{\tan A-\tan B}{1+\tan A\tan B}$$

Since $\tan \left(\frac{\pi }{4}\right)=1$:

$$\tan \left(\frac{\pi }{4}+\theta \right)=\frac{1+\tan \theta }{1-\tan \theta }$$

$$\tan \left(\frac{\pi }{4}-\theta \right)=\frac{1-\tan \theta }{1+\tan \theta }$$

Step 3: Multiply the terms together

Substitute these fractional forms into our modified expression $E$:

$$E=-\left(\frac{1+\tan \theta }{1-\tan \theta }\right)\cdot \left(\frac{1-\tan \theta }{1+\tan \theta }\right)$$

Canceling out the common terms in the numerator and denominator:

$$E=-1$$

Final Answer

The correct option is D (-1).