Hi shashank pi is the first answer that comes to mind but it is not the answer. although the function repeats itself after every pi. But, by definition period is the smallest value for which the function repeats itself. To solve such questions , we can follow 2 approaches : Graphical and Numerical. This is a simple graph and can be done , but it is not always possible to draw graphs. So a more general solution : f(x + T ) = f(x) Case 1 : 0 < x < pi/2 and 0 < x +T < pi/2 sin(x+T) + cos(x+T) = sinx + cosx Multiplying and dividing by sqrt2 sin(x + T + pi/4) = sin(x + pi/4) T = 0 , 2pi , 4pi No solution
Case 2 : 0<x<pi/2 and pi/2 < x+T <pi sin(x+T) - cos(x+T) = sinx + cosx [ Look at the -ve sign] Multiplying and dividing by sqrt2 sin(x + T - pi/4) = sin(x + pi/4) T = pi/2 Thus , pi/2 is the solution. Try drawing the graph and you will understand more clearly !!