Trigonometric Calculation for $cosec^2A$
The question asks for the value of $cosec^2A$ given the expression for $cotA$. We can solve this using a standard trigonometric identity.
Key Trigonometric Identity
The fundamental relationship between $cotA$ and $cosec^2A$ is given by the identity:
$cosec^2A = 1 + cot^2A$
Step-by-Step Solution
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We are given $cotA = x + \frac{1}{x}$
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First, find $cot^2A$ by squaring the given expression for $cotA$:$cot^2A = \left(x + \frac{1}{x}\right)^2$
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Expand the squared term using the formula $(a+b)^2 = a^2 + 2ab + b^2$:$cot^2A = x^2 + 2(x)\left(\frac{1}{x}\right) + \left(\frac{1}{x}\right)^2$$cot^2A = x^2 + 2 + \frac{1}{x^2}$
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Now, substitute this value of $cot^2A$ back into the trigonometric identity $cosec^2A = 1 + cot^2A$:$cosec^2A = 1 + \left(x^2 + 2 + \frac{1}{x^2}\right)$
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Simplify the expression:$cosec^2A = x^2 + \frac{1}{x^2} + 1 + 2$$cosec^2A = x^2 + \frac{1}{x^2} + 3$
Therefore, the value of $cosec^2A$ is $x^2 + \frac{1}{x^2} + 3$. This matches Option 1.
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