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

Simplify to a single trig function with no denominator.\newlinecsc⁡2θ⋅sin⁡2θ \csc ^{2} \theta \cdot \sin ^{2} \theta \newlineAnswer:

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Q. Simplify to a single trig function with no denominator.\newlinecsc⁡2θ⋅sin⁡2θ \csc ^{2} \theta \cdot \sin ^{2} \theta \newlineAnswer:
  1. Understand Trigonometric Identities: Understand the trigonometric identities involved.\newlineThe cosecant function is the reciprocal of the sine function, so csc⁡(θ)=1sin⁡(θ)\csc(\theta) = \frac{1}{\sin(\theta)}. Therefore, csc⁡2(θ)=1sin⁡2(θ)\csc^{2}(\theta) = \frac{1}{\sin^{2}(\theta)}.
  2. Multiply by sin⁡2(θ)\sin^{2}(\theta): Multiply csc⁡2(θ)\csc^{2}(\theta) by sin⁡2(θ)\sin^{2}(\theta). Since csc⁡2(θ)=1sin⁡2(θ)\csc^{2}(\theta) = \frac{1}{\sin^{2}(\theta)}, we have: csc⁡2(θ)×sin⁡2(θ)=(1sin⁡2(θ))×sin⁡2(θ)\csc^{2}(\theta) \times \sin^{2}(\theta) = \left(\frac{1}{\sin^{2}(\theta)}\right) \times \sin^{2}(\theta).
  3. Simplify the Expression: Simplify the expression.\newlineThe sin⁡2(θ)\sin^{2}(\theta) in the numerator and denominator cancel each other out, leaving us with:\newline1sin⁡2(θ)×sin⁡2(θ)=1\frac{1}{\sin^{2}(\theta)} \times \sin^{2}(\theta) = 1.

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