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

Simplify to a single trig function with no denominator.\newlinecot⁡θcos⁡θ \frac{\cot \theta}{\cos \theta} \newlineAnswer:

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Full solution

Q. Simplify to a single trig function with no denominator.\newlinecot⁡θcos⁡θ \frac{\cot \theta}{\cos \theta} \newlineAnswer:
  1. Express cot⁡(θ)\cot(\theta): Express cot⁡(θ)\cot(\theta) in terms of cos⁡(θ)\cos(\theta) and sin⁡(θ)\sin(\theta) as cot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}.
  2. Substitute cot⁡(θ)\cot(\theta): Substitute cot⁡(θ)\cot(\theta) with cos⁡(θ)sin⁡(θ)\frac{\cos(\theta)}{\sin(\theta)} in the given expression to get cos⁡(θ)sin⁡(θ)cos⁡(θ)\frac{\frac{\cos(\theta)}{\sin(\theta)}}{\cos(\theta)}.
  3. Simplify the expression: Simplify the expression by dividing cos⁡(θ)\cos(\theta) by cos⁡(θ)\cos(\theta), which results in 1sin⁡(θ)\frac{1}{\sin(\theta)}.
  4. Recognize the definition: Recognize that 1sin⁡(θ)\frac{1}{\sin(\theta)} is the definition of csc⁡(θ)\csc(\theta), so the expression simplifies to csc⁡(θ)\csc(\theta).

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