Find the general solution of each of the following equations:

(i) cos 3x = cos 2x

(ii) cos 5x = sin 3x

(iii) cos mx = sin nx

To Find: General solution.

(i) Given: cos 3x = cos 2x cos 3x - cos 2x = 0 -2sin sin = 0

[NOTE: cos C – cos D = -2sin sin ]

So, sin = 0 or sin= 0

Formula used: sin = 0 = n , n I

= n or = m where n, m I

x = 2 n/5 or x = 2m where n, m I

So general solution is x = 2 n/5 or x = 2m where n, m I

(ii) Given: cos 5x = sin 3x cos 5x = cos

Formula used: cos = cos = 2n , n I

By using the above formula, we have

5x = 2n or 5x = 2n

8x = 2n or 2x = 2n

x = or x = n where n I

So general solution is x = or x = n where n I

(iii) Given: cos mx = sin nx cos mx = cos

Formula used: cos = cos = 2k , k I

By using the above formula, we have

mx = 2k or 5x = 2k

(m+n)x = 2k or (m-n)x = 2k

x = or x = where k I

x = or x = where k I

So the general solution is x = or x = where k I