Tan theta4Squareroot 30 2 sin 3 theta10 3 sin2 theta10 sin2t

Tan (theta/4)+Squareroot 3=0 2 sin (3 theta)+1=0 3 sin(2 theta)+1=0 sin^2(theta)

a) tan(x/4) +sqrt3 =0

tan(x/4) = -sqrt3

tanx is -ve in IInd quadrant , and IVrt quadrant

x/4 = pi -pi/3 = 2pi/3

General solution : x/4 = k*pi +2pi/3

x = (8pi +12pi*k)/3

b) 2sin(3x) + 1 =0

sin(3x) = -1/2

General solution :3x = 7pi/6 +2k*pi

x =(7pi +12kpi)/18

3x = 11pi/6 +2k*pi

x =( 11pi + 12kpi)/18

c) 3sin3x -sinx =0

3*2sinxcosx -sinx =0

sinx( 6cosx - 1) =0

sinx = 0

General solution: x = 2kpi

cosx = 1/6

General solution:

x = 2k*pi - arcsin(1/6)

x = 2k*pi + arcsin(1/6)

x = 2k*pi +1.40335 ; x = 2k*pi - 1.40335

d) sin^2x = 4 -2cos^2x

sin^2x + 2cos^2x =4

sin^2x +cos^2x +cos^2x = 4

1 +cos^2x = 4

cos^2x = 3

cos2x = sqrt3 >1

As cosx is >1 .

So, no solution exists

2) cos^4x in terms of cosx

cos4x = 2cos^2(2x) -1

and we know cos2x = 2cos^2x -1

cos4x = 2cos^2(2x) -1

= 2 (2cos^2(x) - 1)^2 -1

= 2( 4cos^4x +1 - 4cos^2x) -1

= 8cos^4x +2 - 4cos^2x -1
= 8cos^4x - 4cos^2x + 1