Explained: "Simplify 1−sin⁡(x)+cos⁡(x)1+sin⁡(x)+cos⁡(x)"

chingli20013a1

Answered question

2022-04-27

Simplify

\frac{1 - \sin (x) + \cos (x)}{1 + \sin (x) + \cos (x)}

Tyler Velasquez

Beginner2022-04-28Added 19 answers

\frac{1 - \sin x + \cos x}{1 + \sin x + \cos x} = \frac{\sin^{2} {(\frac{x}{2}) + \cos^{2} {(\frac{x}{2})} - 2 \sin {(\frac{x}{2})} \cos {(\frac{x}{2})} + \cos^{2} {(\frac{x}{2})} - \sin^{2} {(\frac{x}{2})}}}{\sin^{2} {(\frac{x}{2}) + \cos^{2} {(\frac{x}{2})} + 2 \sin {(\frac{x}{2})} \cos {(\frac{x}{2})} + \cos^{2} {(\frac{x}{2})} - \sin^{2} {(\frac{x}{2})}}} =

= \frac{2 \cos^{2} {(\frac{x}{2})} - 2 \sin {(\frac{x}{2})} \cos {(\frac{x}{2})}}{2 \cos^{2} {(\frac{x}{2})} + 2 \sin {(\frac{x}{2})} \cos {(\frac{x}{2})}} =

= \frac{\cos {(\frac{x}{2})} - \sin {(\frac{x}{2})}}{\cos {(\frac{x}{2})} + \sin {(\frac{x}{2})}} = \frac{{(\cos {(\frac{x}{2})} - \sin {(\frac{x}{2})})}^{2}}{(\cos {(\frac{x}{2})} + \sin {(\frac{x}{2})}) (\cos {(\frac{x}{2})} + \sin {(\frac{x}{2})})} =

= \frac{1 - \sin {x}}{\cos {x}}

narratz5dz

Beginner2022-04-29Added 13 answers

\cos 2 y = 1 - 2 \sin^{2} y = 2 \cos^{2} y - 1, \sin 2 y = 2 \sin y \cos y

\frac{1 - \sin 2 y + \cos 2 y}{1 + \sin 2 y + \cos 2 y} = \frac{2 \cos^{2} y - 2 \sin y \cos y}{2 \cos^{2} y + 2 \sin y \cos y} = \frac{\cos y - \sin y}{\cos y + \sin y}

(i) = \frac{{(\cos y - \sin y)}^{2}}{\cos^{2} y - \sin^{2} y} = \frac{1 - \sin 2 y}{\cos 2 y}

(i i) = \frac{\cos^{2} y - \sin^{2} y}{{(\cos y + \sin y)}^{2}} = \frac{\cos 2 y}{1 + \sin 2 y}

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