Simplify the expression: $\sqrt{\frac{1-\sin heta}{1+\sin heta}}$

Question

Simplify the expression: $\sqrt{\frac{1-\sin 	heta}{1+\sin 	heta}}$

Examoogle · Accepted Answer

The correct answer is Option A: $\sec 	heta - 	an 	heta$. Plug in a simple value like $	heta = 0^\circ$. The expression becomes $\sqrt{1/1} = 1$. Evaluating the options at $	heta = 0^\circ$: (A) $\sec 0 - 	an 0 = 1 - 0 = 1$. This matches.

Step-by-Step Solution & Concepts:
Shortcut Trick: Plug in a simple value like $	heta = 0^\circ$. The expression becomes $\sqrt{1/1} = 1$. Evaluating the options at $	heta = 0^\circ$: (A) $\sec 0 - 	an 0 = 1 - 0 = 1$. This matches.

Formula Used: $$\sqrt{\frac{1-\sin 	heta}{1+\sin 	heta}} = \frac{1-\sin 	heta}{\cos 	heta}$$

Given: An expression containing a square root of a ratio involving sine functions: $\sqrt{\frac{1-\sin 	heta}{1+\sin 	heta}}$

Steps: Rationalize the denominator: Multiply the numerator and denominator inside the square root by the conjugate $(1-\sin 	heta)$.. Use trigonometric identity: Apply the identity $1 - \sin^2 	heta = \cos^2 	heta$ to simplify the denominator.. Simplify and separate: Divide each term in the numerator by the denominator $\cos 	heta$ using the definitions of $\sec 	heta$ and $	an 	heta$.

Answer: A is correct because rationalizing the denominator and applying the identity $\cos^2 	heta = 1 - \sin^2 	heta$ results in $\sec 	heta - 	an 	heta$.

Exam Tip: The technique of multiplying by the conjugate is frequently used in limit problems involving square roots in Calculus and in simplifying complex surds in Algebra.

Examoogle · Answer

Option B: $\sec 	heta + 	an 	heta$

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Option C: $	an 	heta - \sec 	heta$

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Option D: $\cos 	heta - \sin 	heta$

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