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(sin theta)/(cot^(2)theta)-(sin theta)/(cos^(2)theta)

sin⁡θcot⁡2θ−sin⁡θcos⁡2θ \frac{\sin \theta}{\cot ^{2} \theta}-\frac{\sin \theta}{\cos ^{2} \theta}

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Full solution

Q. sin⁡θcot⁡2θ−sin⁡θcos⁡2θ \frac{\sin \theta}{\cot ^{2} \theta}-\frac{\sin \theta}{\cos ^{2} \theta}
  1. Express cot⁡(θ)\cot(\theta): Express cot⁡(θ)\cot(\theta) in terms of cos⁡(θ)\cos(\theta) and sin⁡(θ)\sin(\theta), 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 first term to get sin⁡(θ)(cos⁡(θ)sin⁡(θ))2\frac{\sin(\theta)}{(\frac{\cos(\theta)}{\sin(\theta)})^2}.
  3. Simplify first term: Simplify the first term by squaring cos⁡(θ)sin⁡(θ)\frac{\cos(\theta)}{\sin(\theta)} to get sin⁡(θ)cos⁡2(θ)/sin⁡2(θ)\frac{\sin(\theta)}{\cos^{2}(\theta)/\sin^{2}(\theta)}.
  4. Combine terms: Combine the terms over a common denominator to get (sin⁡θ⋅sin⁡2θ)/cos⁡2θ−(sin⁡θ)/cos⁡2θ(\sin \theta \cdot \sin^{2}\theta)/\cos^{2}\theta - (\sin \theta)/\cos^{2}\theta.
  5. Simplify expression: Simplify the expression to get sin⁡3θcos⁡2θ−sin⁡θcos⁡2θ\frac{\sin^{3}\theta}{\cos^{2}\theta} - \frac{\sin \theta}{\cos^{2}\theta}.
  6. Factor out sin⁡θ\sin \theta: Factor out sin⁡θcos⁡2θ\frac{\sin \theta}{\cos^{2}\theta} to get sin⁡θcos⁡2θ×(sin⁡2θ−1)\frac{\sin \theta}{\cos^{2}\theta} \times (\sin^{2}\theta - 1).
  7. Use Pythagorean identity: Use the Pythagorean identity sin⁡2θ+cos⁡2θ=1\sin^2\theta + \cos^2\theta = 1 to rewrite sin⁡2θ−1\sin^2\theta - 1 as −cos⁡2θ-\cos^2\theta.
  8. Substitute sin⁡2θ\sin^2\theta: Substitute sin⁡2θ−1\sin^2\theta - 1 with −cos⁡2θ-\cos^2\theta in the expression to get sin⁡θcos⁡2θ×(−cos⁡2θ)\frac{\sin \theta}{\cos^2\theta} \times (-\cos^2\theta).
  9. Simplify expression: Simplify the expression to get −sin⁡θ⋅cos⁡2θcos⁡2θ-\sin \theta \cdot \frac{\cos^{2}\theta}{\cos^{2}\theta}.
  10. Cancel out cos⁡2θ\cos^2\theta: Cancel out the cos⁡2θ\cos^2\theta terms to get −sin⁡θ-\sin \theta.

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