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Simplify the trigonometric expression \newlinesin⁡x+cot⁡xcos⁡x\sin x + \cot x \cos x

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Full solution

Q. Simplify the trigonometric expression \newlinesin⁡x+cot⁡xcos⁡x\sin x + \cot x \cos x
  1. Recognize Identities: Recognize the trigonometric identities involved.\newlineThe expression contains sin⁡x\sin x, cot⁡x\cot x, and cos⁡x\cos x. We know that cot⁡x\cot x is the reciprocal of tan⁡x\tan x, which means cot⁡x=1tan⁡x\cot x = \frac{1}{\tan x} or cot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}.
  2. Substitute Expression: Substitute the expression for cot⁡x\cot x in terms of sin⁡x\sin x and cos⁡x\cos x.sin⁡x+cot⁡xcos⁡x=sin⁡x+(cos⁡xsin⁡x)cos⁡x\sin x + \cot x \cos x = \sin x + \left(\frac{\cos x}{\sin x}\right) \cos x
  3. Simplify by Multiplying: Simplify the expression by multiplying cos⁡x\cos x with cos⁡xsin⁡x\frac{\cos x}{\sin x}.sin⁡x+(cos⁡xsin⁡x)cos⁡x=sin⁡x+(cos⁡2xsin⁡x)\sin x + \left(\frac{\cos x}{\sin x}\right) \cos x = \sin x + \left(\frac{\cos^2 x}{\sin x}\right)
  4. Combine Terms: Combine the terms over a common denominator.\newlinesin⁡x+cos⁡2xsin⁡x\sin x + \frac{\cos^2 x}{\sin x} \newline= sin⁡2x+cos⁡2xsin⁡x\frac{\sin^2 x + \cos^2 x}{\sin x}
  5. Recognize Another Identity: Recognize another trigonometric identity.\newlineWe know that sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1, which is the Pythagorean identity for sine and cosine.
  6. Apply Pythagorean Identity: Apply the Pythagorean identity to the expression. \newline(sin⁡2x+cos⁡2x)/sin⁡x=1/sin⁡x(\sin^2 x + \cos^2 x)/\sin x = 1/\sin x
  7. Recognize Reciprocal: Recognize that 1sin⁡x\frac{1}{\sin x} is the reciprocal of sin⁡x\sin x, which is csc⁡x\csc x.\newline1sin⁡x=csc⁡x\frac{1}{\sin x} = \csc x
  8. Write Final Expression: Write the final simplified expression.\newlineThe simplified expression is csc⁡x\csc x.

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