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cot x+(sin x)/(1+cos x)=csc x:

cot⁡x+sin⁡x1+cos⁡x=csc⁡x: \cot x+\frac{\sin x}{1+\cos x}=\csc x:

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Csc, sec, and cot of special anglesright chevron icon

Full solution

Q. cot⁡x+sin⁡x1+cos⁡x=csc⁡x: \cot x+\frac{\sin x}{1+\cos x}=\csc x:
  1. Write cot⁡x\cot x in terms: To start, let's write cot⁡x\cot x in terms of sin⁡x\sin x and cos⁡x\cos x.\newlinecot⁡x=cos⁡xsin⁡x\cot x = \frac{\cos x}{\sin x}
  2. Find common denominator: Now, let's focus on the second term (sin⁡x)/(1+cos⁡x)(\sin x)/(1 + \cos x) and find a common denominator with cot⁡x\cot x.\newline(sin⁡x)/(1+cos⁡x)=(sin⁡x)/(1+cos⁡x)⋅(sin⁡x/sin⁡x)(\sin x)/(1 + \cos x) = (\sin x)/(1 + \cos x) \cdot (\sin x/\sin x)
  3. Simplify expression: Simplify the expression by multiplying the numerator and the denominator by sin⁡x\sin x.sin⁡x1+cos⁡x=sin⁡2xsin⁡x⋅(1+cos⁡x)\frac{\sin x}{1 + \cos x} = \frac{\sin^2 x}{\sin x \cdot (1 + \cos x)}
  4. Combine two terms: Combine the two terms cot⁡x\cot x and sin⁡x1+cos⁡x\frac{\sin x}{1 + \cos x}.cot⁡x+sin⁡x1+cos⁡x=cos⁡xsin⁡x+sin⁡2xsin⁡x⋅(1+cos⁡x)\cot x + \frac{\sin x}{1 + \cos x} = \frac{\cos x}{\sin x} + \frac{\sin^2 x}{\sin x \cdot (1 + \cos x)}
  5. Find common denominator: Find a common denominator for the two terms.\newline(cos⁡xsin⁡x)+(sin⁡2xsin⁡x⋅(1+cos⁡x))=(cos⁡x⋅(1+cos⁡x)+sin⁡2xsin⁡x⋅(1+cos⁡x))(\frac{\cos x}{\sin x}) + (\frac{\sin^2 x}{\sin x \cdot (1 + \cos x)}) = (\frac{\cos x \cdot (1 + \cos x) + \sin^2 x}{\sin x \cdot (1 + \cos x)})
  6. Simplify numerator: Simplify the numerator using the Pythagorean identity sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1.cos⁡x⋅(1+cos⁡x)+sin⁡2x\cos x \cdot (1 + \cos x) + \sin^2 x = cos⁡x+cos⁡2x+sin⁡2x\cos x + \cos^2 x + \sin^2 x = cos⁡x+1\cos x + 1

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