Find lim_(theta rarr(pi)/(2))(sin^(2)(2theta))/(1-sin^(2)(theta)) Choose 1 answer: (A) 1 (B) 2 (C) 4 (D) The limit doesn't exist

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Find  lim_(theta rarr(pi)/(2))(sin^(2)(2theta))/(1-sin^(2)(theta)) Choose 1 answer: (A) 1 (B) 2 (C) 4 (D) The limit doesn't exist

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Recognize Limit Issue: First, let's recognize that as theta approaches pi/2, sin(theta) approaches 1. This means that 1 - sin^2(theta) approaches 0, which could lead to a division by zero issue. We need to find a way to simplify the expression to avoid this.Trigonometric Identity: We can use the trigonometric identity sin^2(x) = 1 - cos^2(x) to rewrite the denominator. This gives us:  lim_(theta rarr(pi)/(2))(sin^(2)(2theta))/(1-sin^(2)(theta)) = lim_(theta rarr(pi)/(2))(sin^(2)(2theta))/cos^2(theta)Double Angle Formula: Next, we can use the double angle formula for sine, which is sin(2x) = 2sin(x)cos(x). Applying this to sin^2(2theta), we get:  sin^2(2theta) = (2sin(theta)cos(theta))^2 = 4sin^2(theta)cos^2(theta)Substitute and Simplify: Substituting this back into our limit expression, we have:  lim_(theta rarr(pi)/(2))(4sin^2(theta)cos^2(theta))/cos^2(theta)Cancel Terms: We can now cancel out the cos^2(theta) terms in the numerator and the denominator, which simplifies our expression to:  lim_(theta rarr(pi)/(2))4sin^2(theta)Evaluate Limit: As theta approaches pi/2, sin(theta) approaches 1. Therefore, sin^2(theta) approaches 1^2, which is 1. So our limit becomes:  lim_(theta rarr(pi)/(2))4sin^2(theta) = 4 * 1 = 4Final Answer: The final answer is that the limit of sin^2(2theta)/(1-sin^2(theta)) as theta approaches pi/2 is 4. Therefore, the correct choice is (C) 4.

Find $\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\sin ^{2}(2 \theta)}{1-\sin ^{2}(\theta)}$ $\newline$ Choose $1$ answer: $\newline$ (A) $1$ $\newline$ (B) $2$ $\newline$ (C) $4$ $\newline$ (D) The limit doesn't exist

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Find $\lim _{\theta \rightarrow \frac{\pi}{2}} \frac{\sin ^{2}(2 \theta)}{1-\sin ^{2}(\theta)}$ $\newline$ Choose $1$ answer: $\newline$ (A) $1$ $\newline$ (B) $2$ $\newline$ (C) $4$ $\newline$ (D) The limit doesn't exist