Question

If (sin A)/(sin B) = √3/2 , (cos A)/(cos B) = √5/2, then find the values of tan A and tan B.

Answer 1

sinA/sinB = root 3/2.
=》sin^2 A/sin^2 B = 3/4.
=》sin^2 A = 3sin^2 B/4 ...(1)
Again,
cosA/cosB = root5/2
=》cos^2 A = 5cos^2 B/4 ...(2)
Adding (1) and (2),
cos^2 A + sin^2 A = 1 = 5cos^2 B/4 + 3sin^2 B/4 = (5cos^2 B + 3sin^2 B)/4.
=》4 = 5cos^2 B + 3sin^2 B = 5(1-sin^2 B) + 3sin^2 B = 5 - 2sin^2 B.
=》sin B = 1/root 2 = sin 45.so, tan B = tan 45 = 1 ...(3)
Also,
cos B = cos 45 = 1/root 2.
Again, cosA/cosB = cosA/(1/root2) = root 5/2.
=》cosA = root5/root 8,
(Now, by using square- identity), we get:
tan A = root 3/ root 5 ....(4).
Adding (3) and (4),
tan A + tan B = 1 + root 3/root 5 = (root 5 + root 3)/root 5