Q)- The period of function f(x) = (sin x + sin 3x + sin 5x + sin 7x)/( cos x + cos 3x + cos 5x + cos 7x) is 1) π/6 2) π/3 3) π/4 4) π/2 Share with your friends Share 6 Manvendra S. answered this Dear Student, Please find below the solution to the asked query: fx=sin x+sin 3x+sin 5x+sin 7xcos x+cos 3x+cos 5x+cos 7xFirst to solve the Numerator Beacuse⇒sinA + sinB = 2sinA + B/2cosA - B/2 ⇒cosA + cosB = 2cosA + B/2cosA - B/2 So ⇒sin7x + sinx = 2sin7x + x/2cos7x - x/2 ⇒sin7x + sinx = 2sin4xcos3x ⇒sin5x+ sin3x = 2sin5x + 3x/2cos5x - 3x/2 ⇒sin5x + sin3x = 2sin4xcosx So put these values in total Numerator⇒sinx + sin3x + sin5x + sin7x = 2sin4xcos3x + 2sin4xcosx ⇒sinx + sin3x + sin5x + sin7x = 2sin4xcos3x + cosx Now use the formula ⇒cos3x + cosx = 2cos3x + x/2cos3x - x/2 ⇒cos3x + cosx = 2cos2xcosx Hence ⇒sinx + sin3x + sin5x + sin7x = 2sin4x2cos2xcosx ⇒sinx + sin3x + sin5x + sin7x = 4cosx cos2x sin4x......equation1 Now solve the Denomenator Similarly as aboveuse the formula cosA - B + cosA+B = 2cosA cosB ⇒cos x+cos 3x+cos 5x+cos 7x ⇒cos x + cos7x + cos3x + cos5x ⇒cos4x - 3x + cos4x + 3x + cos4x - x + cos4x + x ⇒2 cos4x cos3x + 2cos4x cosx Using the formula ⇒2 cos4x cos3x + cosx = 2 cos4xcos2x + x+ cos2x - x ⇒2cos4x 2cos2x cosx Using the formula ⇒4 cos4x cos2x cosx ⇒4cosxcos2xcos.....equation2Now put both values in the real functionfx=sin x+sin 3x+sin 5x+sin 7xcos x+cos 3x+cos 5x+cos 7xPut the values of equation 1 and 2 in here fxfx=4cos(x)cos(2x)sin(4x)4cosxcos2xcos4x=tan4xNow we know that the period of tan x is πSo the period of tan 4x will be π4 AnswerHope this information will clear your doubts about the topic. If you have any more doubts just ask here on the forum and our experts will try to help you out as soon as possible. Regards 2 View Full Answer
