The value of $\sin^2 heta + \cos^2 heta - (\sec^2 heta - an^2 heta) + \cot heta \sin heta - \cos

Question

The value of $\sin^2 	heta + \cos^2 	heta - (\sec^2 	heta - 	an^2 	heta) + \cot 	heta \sin 	heta - \cos 	heta $ is ________. ($	heta $ lies in first quadrant)

Examoogle · Accepted Answer

The correct answer is Option B: $0$. Use the standard identities: $\sin^2 	heta + \cos^2 	heta = 1$ and $\sec^2 	heta - 	an^2 	heta = 1$. Substituting these simplifies the start of the expression to $1 - 1 = 0$, leaving only $\cot 	heta \sin 	heta - \cos 	heta = 0$.

Step-by-Step Solution & Concepts:
Shortcut Trick: Use the standard identities: $\sin^2 	heta + \cos^2 	heta = 1$ and $\sec^2 	heta - 	an^2 	heta = 1$. Substituting these simplifies the start of the expression to $1 - 1 = 0$, leaving only $\cot 	heta \sin 	heta - \cos 	heta = 0$.

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

Given: Expression: sin²θ + cos²θ - (sec²θ - tan²θ) + cotθ sinθ - cosθ, where θ is in the first quadrant.

Steps: Apply Pythagorean identities: Substitute the basic trigonometric identities $\sin^2 	heta + \cos^2 	heta = 1$ and $\sec^2 	heta - 	an^2 	heta = 1$ into the given expression.. Simplify cotangent term: Convert $\cot 	heta$ to its sine/cosine form: $\cot 	heta = \frac{\cos 	heta}{\sin 	heta}$. Therefore, $\cot 	heta \sin 	heta = \frac{\cos 	heta}{\sin 	heta} \cdot \sin 	heta = \cos 	heta$.. Final calculation: Combine the simplified terms to reach the final result.

Answer: B is correct because the entire expression simplifies to zero using standard trigonometric identities.

Exam Tip: This simplification technique is fundamental in calculus for solving integration problems involving trigonometric substitution.

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