tan(p/4 + x/2) = sec(x) + tan(x)

Question

I need to prove the above mentioned equation

Answer 1

tan(x+y)=(tan(x)+tan(y))/(1-tan(x)tan(y))

 

tan(pi/4+x/2)=(tan(pi/4)+tan(x/2))/(1-tan(pi/4)tan(x/2))=(1+tan(x/2))/(1-tan(x/2)). There are many different ways to show that the right hand side is equivalent to sec(x)+tan(x). I show a proof using well known double angle formulas.

 

 

Rewrite (1+tan(x/2))/(1-tan(x/2)) as (1+(sin(x/2)/cos(x/2)))/(1-(sin(x/2)/cos(x/2))).

 

Now multiply top and bottom of (1+(sin(x/2)/cos(x/2)))/(1-(sin(x/2)/cos(x/2))) by cos(x/2). Then you will get

 

(cos(x/2)+sin(x/2))/((cos(x/2)-sin(x/2)).

 

Again multiply top and bottom of (cos(x/2)+sin(x/2))/((cos(x/2)-sin(x/2)) by (cos(x/2)+sin(x/2)) to get (cos^2(x/2)+sin^2(x/2)+2cos(x/2)(sin(x/2))/(cos^2(x/2)-sin^2(x/2)).

 

Now  cos^2(x/2)+sin^2(x/2)=1 and we know by well known double angle formulas that

2cos(x/2)(sin(x/2))=sin(x) and (cos^2(x/2)-sin^2(x/2))=cos(x).

 

So (cos^2(x/2)+sin^2(x/2)+2cos(x/2)(sin(x/2))/(cos^2(x/2)-sin^2(x/2))=(1+sin(x))/(cos(x)=sec(x)+tan(x)

 

This completes the proof.

 

 

 

 

 

 

Answer 2

Convert the tangent on the left side of the equation the corresponding sine and cosine values, expand those, and simplify.

 

tan( pi/4 + x/2 )  =  sin( pi/4 + x/2 ) / cos( pi/4 + x/2 )

 

sin ( pi/4 + x/2 )   =   sin( pi/4 ) * cos( x/2 )  +  sin( x/2 ) * cos( pi/4 )

 

cos( pi/4 + x/2 )   =   cos( pi/4 ) * cos( x/2 )  -  sin( pi/4 ) * sin( x/2 )

 

cos( pi/4 )   =   √(2) / 2,  and sin( pi/4 )  =   √(2) / 2

 

tan( pi/4 + x/2 )  =  ( √(2)/2 * (cos( x/2 ) + sin( x/2 )) ) / ( √(2)/2 * (cos( x/2 ) - sin( x/2 )) )

 

the  √(2)/2  values cancel out each other, so

 

tan( pi/4 + x/2 ) = (cos( x/2 ) + sin( x/2 )) / (cos( x/2 ) - sin( x/2 ))

 

now we multiply the numerator and the denominator by  (cos( x/2 ) + sin( x/2 ))

 

this will give us

 

                                 (cos²(x/2) + 2*sin(x/2)*cos(x/2) + sin²(x/2)) 

tan( pi/4 + x/2 ) =    ------------------------------------------------------

                                                    cos²(x/2) - sin²(x/2)

cos²(x/2) + sin²(x/2)   =   1, 

 

2*sin(x/2)*cos(x/2)     =    sin(x), and 

 

cos²(x/2) - sin²(x/2)    =    cos(x), so we have

 

                                    1  +   sin(x)              1                sin(x) 
tan( pi/4 + x/2 )    =    -----------------   =  ---------   +   ---------   =   sec(x)  +  tan(x)
                                         cos(x)               cos(x)           cos(x)