Question

Please help me solve this problem

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

Thank you in advance!!

Answer 1

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