Question

if cos-sin=1, then show that cos+sin=1 or -1

Answer 1

Dear Student,


$$\begin{gathered} let\cos x+\sin x=y \\ Then \\ {\left(\cos x-\sin x\right)}^2+{\left(\cos x+\sin x\right)}^2 \\ {\cos }^{2}x+{\sin }^{2}x-2\cos x\sin x+{\cos }^{2}x+{\sin }^{2}x+2\cos x\sin x \\ 2{\cos }^{2}x+2{\sin }^{2}x \\ =2\left({\cos }^{2}x+{\sin }^{2}x\right)--(as{\cos }^{2}x+{\sin }^{2}x=1) \\ =2 \\ Hence{\left(\cos x-\sin x\right)}^2+{\left(\cos x+\sin x\right)}^2=2 \\ giventhat\cos x-\sin x=1and\cos x+\sin x=y \\ {1}^2+y^2=2 \\ \Rightarrow y^2=2-1 \\ \Rightarrow y^2=1 \\ \Rightarrow y=\pm 1 \\ Hence\cos x+\sin x=\pm 1 \end{gathered}$$

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