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Simplify to a single trig function with no denominator. cot theta*sin theta Answer: theta

Simplify to a single trig function with no denominator.\newlinecot⁡θ⋅sin⁡θ \cot \theta \cdot \sin \theta \newlineAnswer:

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Full solution

Q. Simplify to a single trig function with no denominator.\newlinecot⁡θ⋅sin⁡θ \cot \theta \cdot \sin \theta \newlineAnswer:
  1. Recall Definition of Cotangent: To simplify the expression cot⁡(θ)sin⁡(θ)\cot(\theta)\sin(\theta), we need to recall the definition of cotangent in terms of sine and cosine.\newlineThe cotangent of an angle is the cosine of that angle divided by the sine of that angle, which can be written as:\newlinecot⁡(θ)=cos⁡(θ)sin⁡(θ)\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\newlineNow, we can substitute this definition into our original expression.
  2. Substitute Definition: After substituting the definition, we get:\newlinecot⁡(θ)sin⁡(θ)=(cos⁡(θ)sin⁡(θ))sin⁡(θ)\cot(\theta)\sin(\theta) = \left(\frac{\cos(\theta)}{\sin(\theta)}\right) \sin(\theta)\newlineNow, we can simplify the expression by canceling out the sin⁡(θ)\sin(\theta) in the numerator and the denominator.
  3. Simplify Expression: After canceling out sin⁡(θ)\sin(\theta), we are left with: cos⁡(θ)sin⁡(θ)×sin⁡(θ)=cos⁡(θ)\frac{\cos(\theta)}{\sin(\theta)} \times \sin(\theta) = \cos(\theta) So, the simplified expression is just cos⁡(θ)\cos(\theta).

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