The quotient rule gives
y ′ = − sin x ( sin x + 1 ) − cos x cos x ( sin x + 1 ) 2 = − sin x + ( sin 2 x + cos 2 x ) ( sin x + 1 ) 2 = − sin x + 1 ( sin x + 1 ) 2 = − 1 sin x + 1 {\displaystyle y'={\frac {-\sin x(\sin x+1)-\cos x\cos x}{(\sin x+1)^{2}}}=-{\frac {\sin x+(\sin ^{2}x+\cos ^{2}x)}{(\sin x+1)^{2}}}=-{\frac {\sin x+1}{(\sin x+1)^{2}}}=-{\frac {1}{\sin x+1}}} .
Where the third inequality follows from the Pythagorean identity sin 2 x + cos 2 x = 1 {\displaystyle \sin ^{2}x+\cos ^{2}x=1} for every real number x {\displaystyle x} .
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for every real number 
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