Answer:
To prove that tan A = 12/5, we can use the identity that relates the sine and cosine of an angle to its tangent:
tan A = sin A / cos A
We are given that 0.5 * cos A = 13, so cos A = 26.
Substituting this value into the identity above, we get:
tan A = sin A / 26
We are also given that 12 sec A = 13, so sec A = 13/12 = 1.083333.
The reciprocal of sec A is cosec A, so cosec A = 1/1.083333 = 0.9230769.
The sine of an angle is the reciprocal of the cosecant of the angle, so sin A = 1/0.9230769 = 1.083333.
Substituting this value into the identity tan A = sin A / cos A, we get:
tan A = 1.083333 / 26 = 12/5
Therefore, tan A = 12/5.