(1+tanx)/(sinx+cosx)=secx

[tex]\frac{1+\tan x}{\sin x+\cos x}=\sec x[/tex] is provedSolution:Given that,[tex]\frac{1+\tan x}{\sin x+\cos x}=\sec x[/tex]  ------- (1)First we will simplify the LHS and then compare it with RHS[tex]\text { L. H.S }=\frac{1+\tan x}{\sin x+\cos x}[/tex]  ------ (2)[tex]\text {We know that } \tan x=\frac{\sin x}{\cos x}[/tex]Substitute this in eqn (2)[tex]=\frac{1+\frac{\sin x}{\cos x}}{\sin x+\cos x}[/tex]On simplification we get,[tex]=\frac{\frac{\sin x+\cos x}{\cos x}}{\sin x+\cos x}[/tex][tex]=\frac{\sin x+\cos x}{\cos x} \times \frac{1}{\sin x+\cos x}[/tex]Cancelling the common terms (sinx + cosx)[tex]=\frac{1}{c o s x}[/tex]We know secant is inverse of cosine[tex]=\sec x=R . H . S[/tex]Thus L.H.S = R.H.S[tex]\frac{1+\tan x}{\sin x+\cos x}=\sec x[/tex] Hence proved