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Simplify to a single trig function with no denominator. (cot^(2)theta)/(csc^(2)theta) Answer: theta

Simplify to a single trig function with no denominator.\newlinecot⁡2θcsc⁡2θ \frac{\cot ^{2} \theta}{\csc ^{2} \theta} \newlineAnswer:

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Q. Simplify to a single trig function with no denominator.\newlinecot⁡2θcsc⁡2θ \frac{\cot ^{2} \theta}{\csc ^{2} \theta} \newlineAnswer:
  1. Express cot⁡(θ)\cot(\theta): Express cot⁡(θ)\cot(\theta) and csc⁡(θ)\csc(\theta) in terms of sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta) as cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} and csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}.
  2. Substitute cot⁡(θ)\cot(\theta): Substitute cot⁡(θ)\cot(\theta) and csc⁡(θ)\csc(\theta) in the given expression with their respective expressions in terms of sin⁡(θ)\sin(\theta) and cos⁡(θ)\cos(\theta) to get (cos⁡(θ)sin⁡(θ))2/(1sin⁡(θ))2\left(\frac{\cos(\theta)}{\sin(\theta)}\right)^2 / \left(\frac{1}{\sin(\theta)}\right)^2.
  3. Simplify the expression: Simplify the expression by multiplying the numerator and the denominator by sin⁡(θ)2\sin(\theta)^2 to get cos⁡(θ)2sin⁡(θ)2⋅sin⁡(θ)21\frac{\cos(\theta)^2}{\sin(\theta)^2} \cdot \frac{\sin(\theta)^2}{1}.
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