How do I prove that 1/(sec A+1)+1/(sec A-1)=2 csc A cot A ?

1 Answer
Oct 4, 2015

1 / (sec A + 1) + 1 / (Sec A - 1)

Taking the Lowest common Multiple,

(Sec A - 1 + Sec A + 1) / (Sec A +1) * (Sec A - 1)

As you may be aware, a^2 - b^2 = (a + b) * (a - b)

Simplifying, (2 Sec A) / (Sec^2 A - 1)

Now Sec^2 A - 1 = tan^2 A = Sin^2A / Cos^2A
and Sec A = 1 / Cos A

Substituting,

2 / Cos A * Cos^2A / Sin^2A = 2 * Cos A / Sin^2A

which can be written as 2 * Cos A / Sin A * ( 1 / Sin A)

Now Cos A / Sin A = Cot A and 1 / Sin A = Cosec A
Substituting, we get 2 Cot A * Cosec A