SOLUTION: Prove that 1 + tanAtanA/2 = tanAcotA/2 -1 = secA

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Question 552116: Prove that
1 + tanAtanA/2 = tanAcotA/2 -1 = secA
Answer by AnlytcPhil(1810) About Me  (Show Source):
You can put this solution on YOUR website!
1 + tan(A)·tan%28A%2F2%29 = tan(A)cot%28A%2F2%29 - 1 = sec(A)

We will make use of the half angle identities:

tan%28A%2F2%29 = %281-cos%28A%29%29%2Fsin%28A%29, and its reciprocal cot%28A%2F2%29 = sin%28A%29%2F%281-cos%28A%29%29, also tan(A) = sin%28A%29%2Fcos%28A%29 and sec(A) = 1%2Fcos%28A%29

First we will prove:

1 + tan(A)·tan%28A%2F2%29 = sec(A)

1 + %28sin%28A%29%2Fcos%28A%29%29%28%281-cos%28A%29%29%2Fsin%28A%29%29 

1 + +++%28sin%28A%29-sin%28A%29cos%28A%29%29++%2F%28cos%28A%29sin%28A%29%29+++

Get an LCD

%28cos%28A%29sin%28A%29%29%2F%28cos%28A%29sin%28A%29%29 + +++%28sin%28A%29-sin%28A%29cos%28A%29%29++%2F%28cos%28A%29sin%28A%29%29+++

Combine fractions over the LCD



Cancel first and third terms on top:



sin%28A%29++%2F%28cos%28A%29sin%28A%29%29+++

Cancel the sines:

cross%28sin%28A%29%29++%2F%28cos%28A%29cross%28sin%28A%29%29%29+++

1%2Fcos%28A%29

sec(A)

---------------------------------

Next we will prove:

tan(A)·cot%28A%2F2%29 - 1 = sec(A)

%28sin%28A%29%2Fcos%28A%29%29%28sin%28A%29%2F%281-cos%28A%29%29%29 - 1 

+++%28sin%5E2%28A%29%29++%2F%28cos%28A%29%281-cos%28A%29%29%29+++ - 1

Get an LCD

+++%28sin%5E2%28A%29%29++%2F%28cos%28A%29%281-cos%28A%29%29%29+++ - +++%28cos%28A%29%281-cos%28A%29%29%29++%2F%28cos%28A%29%281-cos%28A%29%29%29+++

Combine fractions over the LCD



Distribute:



Rearrange:



Use identity sin²(A)+cos²(A)=1

+++%281-cos%28A%29%29++%2F%28cos%28A%29%281-cos%28A%29%29%29+++

Cancel 1-cos(A)'s

+++%28cross%281-cos%28A%29%29%29++%2F%28cos%28A%29%28cross%281-cos%28A%29%29%29%29+++

1%2Fcos%28A%29

sec(A)

Edwin