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Prove:

(Sinx + Cosx) (tanx + cotx) = Secx + Cscx


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Verified Answer :

gmany

\((\sin x+\cos x)(\tan x+\cot x)=\sec x+\csc x\\\\L_s=(\sin x+\cos x)\left(\dfrac{\sin x}{\cos x}+\dfrac{\cos x}{\sin x}\right)\\\\=\sin x\cdot\dfrac{\sin x}{\cos x}+\sin x\cdot\dfrac{\cos x}{\sin x}+\cos x\cdot\dfrac{\sin x}{\cos x}+\cos x\cdot\dfrac{\cos x}{\sin x}\\\\=\dfrac{\sin^2x}{\cos x}+\dfrac{\sin x\cos x}{\sin x}+\dfrac{\cos x\sin x}{\cos x}+\dfrac{\cos^2x}{\sin x}\\\\=\dfrac{\sin^3x+\sin x\cos^2x+\cos x\sin^2x+\cos^3x}{\sin x\cos x}\)\(=\dfrac{\sin x(\sin^2x+\cos^2x)+\cos x(\sin^2x+\cos^2x)}{\sin x\cos x}\\\\=\dfrac{\sin x\cdot1+\cos x\cdot1}{\sin x\cos x}=\dfrac{\sin x}{\sin x\cos x}+\dfrac{\cos x}{\sin x\cos x}=\dfrac{1}{\cos x}+\dfrac{1}{\sin x}\\\\=\sec x+\csc x=R_s\)

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