please prove cos A /1- sin A = Tan (45 +_ A/2)trignometry from SL Loney s book

giorgiana1976 | Student

We'll evaluate the right side of expression using the tangent of a sum/difference formula:

tan(a+b) = (tan a + tan b)/(1 - tan a*tan b)

Comparing, we'll get:

tan(45 + A/2) = (tan 45 + tan A/2)/(1 - tan A/2) = (1 +tan A/2)/(1 - tan A/2)

Now, we'll manage the left side,using the double angle identities:

cos A = cos 2(A/2) = (cos A/2)^2 - (sinA/2)^2

The difference of two squares will return the product:

(cos A/2)^2 - (sinA/2)^2 = (cos A/2-sinA/2)(cos A/2+sinA/2)

1 - sin A = 1 - sin 2(A/2)

We'll use Pythagorean identity to replace 1:

(cos A/2)^2 + (sinA/2)^2 = 1

1 - sin 2(A/2) = (cos A/2)^2 + (sinA/2)^2 - 2sin(A/2)*cos (A/2)

We'll get a perfect square at denominator:

1 - sin A = (cos A/2+sinA/2)^2

We'll re-write the left side:

cos A/(1 - sin A) = (cos A/2-sinA/2)(cos A/2+sinA/2)/(cos A/2+sinA/2)^2

We'll simplify:

cos A/(1 - sin A) = (cos A/2-sinA/2)/(cos A/2+sinA/2)

We'll factorize by cos (A/2) both numerator and denominator:

cos A/(1 - sin A) = cos A/2*(1 -sinA/2/cos A/2)/cos A/2*(1+sinA/2/cos A/2)

But sinA/2/cos A/2 = tan A/2

We'll simplify:

cos A/(1 - sin A) = (1 -tan A/22)/(1+tan A/2)

Managing both sides, we notice that only the expression cos A/(1 - sin A) = tan(45 - A/2) represents an identity, while the expression cos A/(1 - sin A) = tan(45 + A/2) is not an identity.