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

Simplify to a single trig function with no denominator.\newlinetan⁡θ⋅cot⁡2θ \tan \theta \cdot \cot ^{2} \theta \newlineAnswer:

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Q. Simplify to a single trig function with no denominator.\newlinetan⁡θ⋅cot⁡2θ \tan \theta \cdot \cot ^{2} \theta \newlineAnswer:
  1. Multiply and Simplify: Now, let's multiply tan⁡(θ)\tan(\theta) by cot⁡2(θ)\cot^2(\theta) using the definitions we've just recalled:\newlinetan⁡(θ)⋅cot⁡2(θ)=(sin⁡(θ)/cos⁡(θ))⋅(cos⁡2(θ)/sin⁡2(θ))\tan(\theta) \cdot \cot^2(\theta) = (\sin(\theta) / \cos(\theta)) \cdot (\cos^2(\theta) / \sin^2(\theta))
  2. Cancel Common Terms: Next, we simplify the expression by canceling out common terms: \newline(sin⁡(θ)/cos⁡(θ))⋅(cos⁡2(θ)/sin⁡2(θ))=sin⁡(θ)⋅cos⁡(θ)/sin⁡2(θ)(\sin(\theta) / \cos(\theta)) \cdot (\cos^2(\theta) / \sin^2(\theta)) = \sin(\theta) \cdot \cos(\theta) / \sin^2(\theta)\newlineHere, cos⁡(θ)\cos(\theta) in the numerator and one sin⁡(θ)\sin(\theta) in the denominator cancel out:\newline=cos⁡(θ)/sin⁡(θ)= \cos(\theta) / \sin(\theta)
  3. Use Definition of cot(theta): The expression cos⁡(θ)sin⁡(θ)\frac{\cos(\theta)}{\sin(\theta)} is the definition of cot⁡(θ)\cot(\theta):cos⁡(θ)sin⁡(θ)=cot⁡(θ)\frac{\cos(\theta)}{\sin(\theta)} = \cot(\theta)So, the simplified expression is cot⁡(θ)\cot(\theta).

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