Question

2 se the double-angle identities to verify the identity 1+cos(2x 2 cos* x = 9. Solve exactly over the indicated interval. a) sin(2x)-cos.x, all real numbers b) 2 cos(29) =-1, 0 θ < 2π

Answer 1

Use double angle identity to verify the identity cos^2x = 1 + cos (2x)/2 R.H.S = 1 + cos (2x)/2 now using the double angle identity i.e., cos (2x) = 2 cos^2 x - 1 R.H.S = 1 + 2 cos^2 x - 1/2 = 2 cos^2 x/2 = cos^2 x = L.H.S hence verified sin (2x) = cos x now using the identity sin 2x = 2 sin x cos x 2 sin x cos x = cos x 2 sin x cos x - cos x = 0 cos x (2 sin x - 1) = 0 cos x = 0 we get x = pi/2, 3pi/2 as cos pi/2 = cos 3pi/2 = 0 i.e., x have the values x = pi/2, 3pi/2, pi/6, 5pi/6 or 2 sin x - 1 = 0 we get sin x = 1/2 x = pi/6m 5pi/6 as sin pi/6 = sin 5pi/6 = 1/2 now for all real no just go with upper values with