Question

tan((pi)/(4)+(x)/(2))+tan((pi)/(4)-(x)/(2))=2 sec x Prove it.

Answer 1

Smple question

we know that tanπ/4 =1

Tan(A+B)={TanA+TanB)/(1-TanATanB)

now using this formula we can write

LHS=(1+Tanx/2)/(1-tanx/2) +(1-tanx/2)/(1+tanx/2)

LHS={(1+tanx/2)^2 +(1-tanx/2)^2}/(1-tan^2x)

LHS=(1+tan^2x/2 +2tanx/2+1+tan^2x/2-2tanx/2)/(sec^2x/2)

LHS=(2+2tan^2x/2)/(sec^2x/2)

LHS=2(1+tan^2x/2)/(sec^2x/2)

=2

LHS=RHS=2