Step by step explanation:
Starting with the given equation:
3 sin A + 4 cos A = 5
We can square both sides:
(3 sin A + 4 cos A)^2 = 5^2
Expanding the left-hand side using the identity (a + b)^2 = a^2 + 2ab + b^2, we get:
9 sin^2 A + 24 sin A cos A + 16 cos^2 A = 25
Using the identity sin^2 A + cos^2 A = 1, we can replace sin^2 A with 1 - cos^2 A, giving:
9(1 - cos^2 A) + 24 sin A cos A + 16 cos^2 A = 25
Simplifying, we get:
9 - 9 cos^2 A + 16 cos^2 A + 24 sin A cos A = 25
Combining like terms, we get:
7 cos^2 A + 24 sin A cos A - 16 = 0
Dividing both sides by cos^2 A (which we assume is not equal to zero), we get:
7 + 24 tan A - 16 sec A = 0
Using the identity tan A = sin A / cos A and sec A = 1 / cos A, we can rewrite this equation as:
7 + 24 (sin A / cos A) - 16 (1 / cos A) = 0
Multiplying both sides by cos A, we get:
7 cos A + 24 sin A - 16 = 0
Now we can solve for tan A:
tan A = sin A / cos A
tan A = (3 sin A) / (4 cos A)
tan A = (3/4) (sin A / cos A)
tan A = 3/4
Therefore, we have proved that if 3 sin A + 4 cos A = 5, then tan A = 3/4.