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

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

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Q. Simplify to a single trig function with no denominator.\newlinecsc⁡2θ⋅cos⁡2θ \csc ^{2} \theta \cdot \cos ^{2} \theta \newlineAnswer:
  1. Recall definition of cosecant function: Recall the definition of the cosecant function. The cosecant function is the reciprocal of the sine function, so csc(θ)=1sin⁡(θ)\text{csc}(\theta) = \frac{1}{\sin(\theta)}.
  2. Rewrite csc⁡2θ\csc^2\theta: Rewrite csc⁡2θ\csc^2\theta as (1/sin⁡(θ))2(1/\sin(\theta))^2. Using the definition from Step 11, we can rewrite csc⁡2θ\csc^2\theta as (1/sin⁡(θ))2(1/\sin(\theta))^2.
  3. Apply power to the fraction: Apply the power to the fraction.\newlineWhen we raise a fraction to a power, we raise both the numerator and the denominator to that power. So, (1/sin⁡(θ))2(1/\sin(\theta))^2 becomes 12/sin⁡2θ1^2/\sin^{2}\theta, which simplifies to 1/sin⁡2θ1/\sin^{2}\theta.
  4. Multiply the expressions: Multiply the expressions.\newlineNow we multiply 1sin⁡2θ\frac{1}{\sin^{2}\theta} by cos⁡2θ\cos^{2}\theta. This gives us (1sin⁡2θ)∗cos⁡2θ\left(\frac{1}{\sin^{2}\theta}\right) * \cos^{2}\theta.
  5. Use Pythagorean identity: Use the Pythagorean identity for sine and cosine.\newlineThe Pythagorean identity states that sin⁡2θ+cos⁡2θ=1\sin^{2}\theta + \cos^{2}\theta = 1. We can rearrange this to express sin⁡2θ\sin^{2}\theta in terms of cos⁡2θ\cos^{2}\theta: sin⁡2θ=1−cos⁡2θ\sin^{2}\theta = 1 - \cos^{2}\theta.
  6. Substitute sin⁡2θ\sin^2\theta: Substitute sin⁡2θ\sin^2\theta in the expression.\newlineSubstitute sin⁡2θ\sin^2\theta with 1−cos⁡2θ1 - \cos^2\theta in the expression from Step 44. This gives us 11−cos⁡2θ×cos⁡2θ\frac{1}{1 - \cos^2\theta} \times \cos^2\theta.
  7. Simplify the expression: Simplify the expression.\newlineSince we have a common factor of cos⁡2θ\cos^{2}\theta in the numerator and the denominator, we can simplify the expression to just cos⁡2θ\cos^{2}\theta. This is because (1/(1−cos⁡2θ))⋅cos⁡2θ=cos⁡2θ/(1−cos⁡2θ)⋅(1−cos⁡2θ)=cos⁡2θ(1/(1 - \cos^{2}\theta)) \cdot \cos^{2}\theta = \cos^{2}\theta / (1 - \cos^{2}\theta) \cdot (1 - \cos^{2}\theta) = \cos^{2}\theta.

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