Question

Tan (x+π/4)= (cosx + sinx)/(cosx-sinx) How do I prove this?  

 

 

Answer 1

Given trigonometric identity : tan (x+π/4) = (cosx + sinx) / (cosx - sinx)

Start from left hand side :

tan (x+π/4)

Apply formula : tan (a+b) = (tana + tanb) / (1 - tana tanb)

= (tanx + tan(π/4) ) / (1 - tanx tan(π/4))

Substitute : tan(π/4) = 1

= (tanx + 1 ) / (1 - tanx (1))

= (tanx + 1 ) / (1 - tanx )

Substitute : tanx = sinx / cosx

= ((sinx/cosx) + 1 ) / (1 - (sinx/cosx))

= ((sinx+cosx)/cosx) / ((cosx-sinx)/cosx)

Both denominators gets canceled.

= (cosx + sinx) / (cosx - sinx)

Hence Trigonometric identity  tan (x+π/4) = (cosx + sinx) / (cosx - sinx) is proved.