Answer: We can start by using the identity sin^2(A) + cos^2(A) = 1.
We can square the equation 3 sin A + 4 cos A = 5, to get:
9sin^2(A) + 12sin(A)cos(A) + 16cos^2(A) = 25
Substituting the identity sin^2(A) + cos^2(A) = 1, we get:
9sin^2(A) + 12sin(A)cos(A) + 16(1-sin^2(A)) = 25
12sin(A)cos(A) = 6
Now,we can use identity tan(A) = sin(A) / cos(A) and sec(A) = 1 / cos(A)
tan(A) = sin(A) / cos(A) = (2sin(A)cos(A)) / (cos^2(A)) = 2 * (sin(A)cos(A)) / (1-sin^2(A)) = 2 * (6 / 12) = 1
sec(A) = 1 / cos(A) = 1 / sqrt(1 - sin^2(A)) = 1 / sqrt(1 - (3/5)^2) = 1 / sqrt(1 - 9/25) = 5/sqrt(16) = 5/4
Now we can add them:
tan(A) + sec(A) = 1 + 5/4 = (4+5) / 4 = 9/4 = 2.25
Hence proved that tan A + sec A = 2.25
Explanation: