The expression $\frac{3 - 4 \sin^2 heta}{\cos^2 heta} + 2 an^2 heta $ can be simplified as:

Question

The expression $\frac{3 - 4 \sin^2 	heta}{\cos^2 	heta} + 2	an^2 	heta $ can be simplified as:

Examoogle · Accepted Answer

The correct answer is Option C: $3 + 	an^2 	heta$. Substitute $	heta = 0^\circ$. The expression becomes $\frac{3 - 0}{1} + 0 = 3$. Checking options: A) 0, B) $3 	an^2 	heta$, C) $3$, D) $3 + 	an^2 	heta$. For $	heta = 0^\circ$, only Option C gives 3.

Step-by-Step Solution & Concepts:
Shortcut Trick: Substitute $	heta = 0^\circ$. The expression becomes $\frac{3 - 0}{1} + 0 = 3$. Checking options: A) 0, B) $3 	an^2 	heta$, C) $3$, D) $3 + 	an^2 	heta$. For $	heta = 0^\circ$, only Option C gives 3.

Formula Used: $$\sin^2 	heta + \cos^2 	heta = 1$$, $$	an 	heta = \frac{\sin 	heta}{\cos 	heta}$$, $$\sec^2 	heta = 1 + 	an^2 	heta$$

Given: Expression: \frac{3 - 4 \sin^2 	heta}{\cos^2 	heta} + 2 	an^2 	heta

Steps: Expand the fraction: Divide the terms in the numerator by the denominator $\cos^2 	heta$.. Use identity for tangent: Apply the identity $	an 	heta = \frac{\sin 	heta}{\cos 	heta}$ and $\sec 	heta = \frac{1}{\cos 	heta}$.. Simplify terms: Combine the $	an^2 	heta$ terms: $-4 	an^2 	heta + 2 	an^2 	heta = -2 	an^2 	heta$.. Apply secant-tangent identity: Use the identity $\sec^2 	heta = 1 + 	an^2 	heta$.

Answer: Option C is correct because the expression simplifies to $3 + 	an^2 	heta$.

Exam Tip: This type of simplification frequently appears in SSC and RRB exams; always check simple angles like $0^\circ$ or $45^\circ$ to verify expressions quickly.

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