of tan π . TRIGONOMETRIC FUNCTIONS Solution We have tan π = tan π π + = tan tan π π π tan tan tan tan π π π π Example Prove that sin( tan tan sin( tan tan Solution We have L.H.S. sin( sin cos cos sin sin( sin cos cos sin Dividing the numerator and denominator by cos x cos y , we get sin( tan tan sin( tan tan Example Show that tan x tan x tan x = tan x – tan x – tan x Solution We know that x = x + x Therefore, tan x = tan ( x + x ) or tan2 tan tan3 – tan tan or tan x – tan x tan x tan x = tan x + tan x or tan x – tan x – tan x = tan x tan x tan x or tan x tan x tan x = tan x – tan x – tan x. Example Prove that cos cos cos π π Solution Using the Identity (i), we have MATHEMATICS L.H.S.
cos cos π π 2cos cos x – π π π π = cos π cos x = × cos x = cos x = R.H.S. Example Prove that cos cos cot sin – sin Solution Using the Identities (i) and (iv), we get L.H.S. = 2cos cos 2cos sin cos sin cot = R.H.S. Example Prove that sin5 2sin3 sin tan cos5 cos Solution We have L.H.S.
sin5 2sin3 sin cos5 cos sin5 sin 2sin3 cos5 cos 2sin3 cos2 2sin3 – 2sin3 sin sin3 (cos2 ) sin3 sin – cos2 2sin sin 2sin cos = tan x = R.H.S. TRIGONOMETRIC FUNCTIONS EXERCISE . Prove that: . sin π + cos