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Question Number 215277 by efronzo1 last updated on 02/Jan/25

$$(1+\cos \mathrm{x})(1+\sin \mathrm{x})=\frac{5}{4}$$$$(1-\cos \mathrm{x})(1-\sin \mathrm{x})=?$$

Answered by MrGaster last updated on 02/Jan/25

$$\underset{\mathrm{Commutative}\mathrm{law}\mathrm{of}\mathrm{multiplication}}{\underbrace{(1+\cos x)(1+\sin x)\cdot (1-\cos x)(1-\sin x)}}=\left(\frac{5}{4}\right)(1-\cos x)(1-\sin x)$$$$(1-{\cos }^{2}x)(1-{\sin }^{2}x)=\left(\frac{5}{4}\right)(1-\cos x)(1-\sin x)$$$$\left({\sin }^{2}x\right)\left({\cos }^{2}x\right)=\left(\frac{5}{4}\right)(1-\cos x)(1-\sin x)$$$$\frac{1}{4}{\sin }^2\left(2x\right)=\left(\frac{5}{4}\right)(1-\cos x)(1-\sin x)$$$$(1-\cos x)(1-\sin x)=\frac{{\sin }^2\left(2x\right)}{5}$$

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