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The value of the expression \( \frac{\sin ^{3} x}{1+\cos x}+\frac{\cos ^{3} x}{1+\sin x} \) is/are :(a) \( \sqrt{2} \cos \left(\frac{\pi}{4}-x\right) \) (b) \( \sqrt{2} \cos \left(\frac{\pi}{4}+x\right) \) (c) \( \sqrt{2} \sin \left(\frac{\pi}{4}-x\right) \) (d) \( \sqrt{2} \sin \left(\frac{\pi}{4}+x\right) \) |
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Answer» \(\frac{sin^3x}{1 + cos x} + \frac{cos^3x}{1 + sin \,x}\) \(= \frac{sin^3x}{1 + cos \,x} \times \frac{1- cos\,x}{1 - cos \,x} + \frac{cos^3x}{1 + sin\,x} \times \frac{1 - sin\,x}{1 - sin\, x}\) \(= \frac{(sin^3x)(1 - cos\,x)}{1 - cos^2x} + \frac{cos^3x}{1 - sin^2x} (1 - sin\,x)\) \(= \frac{(sin^3x)(1 - cos\,x)}{sin^2x} + \frac{cos^3x(1 - sin\,x)}{cos^2x}\) \(= sin\,x (1 - cos\,x) + cos\, x(1 - sin\, x)\) \(= sin\,x + cos\,x - 2sin\,x\;cos\,x\) \(= \sqrt 2\left(\frac1{\sqrt2} sin\,x + \frac1{\sqrt2} cos\,x\right) - sin(2x)\) \(= \sqrt 2 (sin\,x \;cos(\frac\pi4) + cos\,x \;sin(\frac\pi4)) - sin (2x)\) \(= \sqrt 2\; sin(x + \frac\pi4) - sin(2x)\) or \(\sqrt 2\, cos(x - \frac\pi4) - sin(2x)\) ⇒ \(\sqrt 2 \,cos(\frac\pi4 - x) - sin(2x)\) |
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