Verified answer

a)

Cos 2x = Cos^2(x) - Sin^2(x)

&

Sin^2(x) = 1 - Cos^2(x)

Substituting back in

Cos2x = cos^2(x) - (1 - Cos^2(x))

Cos2x = 2Cos^2(x) - 1

Hence

2Cos^2(x) - 1 + 3Cosx = 4

2Cos^2(x) + 3Cos(x) - 5 = 0

It is now in quadratic form

(2Cosx + 5)(Cosx - 1) = 0

Cosx - 1 = 0

Cosx = 1

x = Cos^-1 (1)

x = 0 and 2pi.

Also

2Cosx + 5 = 0

Cosx = - 5/2 This is UNRESOLVED as Cos cannot be greater than '1' or less than '-1'.

b).

Sinx / cosx + Cosx/sinx = 4(2)SinxCosx

(Sin^2x + Cos^2x) / SinxCosx = 8SinxCosx

Sin^2 + Cos^2 = 1

Hence

1 / SinxCosx = 8 SinxCosx

1/8 = (Sin^2x)(Cos^2x)

1/8 = Sin^2x(1 - Sin^2x)

1/8 = Sin^2x - Sin^4x

Sin^4 - Sin^2x + 1/8= 0

Let Sin^2x = y

Hence

y^2 - y + 1/8 = 0

y^2 - y = -1/8

(y - 1/2)^2 -1/4 =-1/8

(y - 1/2)^2 = 1/8

y - 1/2 = +/- 1/2(sqrt2)

y = 1/2 +/- 1/2(sqrt(2)

y = 1/2 +/- sqrt(2)/ 4

y = 1/2 +/- 1.4142/4

y = 0.5 +/- 0.3535...

Sin^2x = 0.8535...

Sinx = sqrt(0.8535...)

x = Sin^-1(+/-sqrt(0.8535...)

x = Sin^-1(+sqrt(0.8535) = 1.178 rads (3pi/8)

x = Sin^-1(-sqrt(0.8535..) = -1.178 rads (-3pi/8)

&

y = 0.1465...

Sin^2 = 0.1465...

Sinx = +/-sqrt(0.1465...) = +/-0.3827...

x = Sin^-1(+0.3827...) = 0.39277..rads (pi/8)

x = Sin^-1(-0.3827...) = -0.3927... rads ( -pi/8)

Since the solution is between 0 & 2pi the negative results can be discounted.

sin2x + sinx = 0 2sinxcosx + sinx = 0 (the double perspective formula grew to become into used to swap the sin2x into 2sinxcosx) sinx(2cosx + one million) = 0 (sinx grew to become into factored out) From there, use the 0 product belongings; it branches this out into 2 issues. the 1st section, sinx = 0. that could desire to alter into pi/2 and 3pi/2. the 2d section, 2cosx+one million = 0: 2cosx= -one million (one million grew to become into subtracted to different ingredient) cosx= -one million/2 (2 grew to become into divied) So the 2d section, cosx= -one million/2. that could desire to alter into 3pi/4 and 5pi/4. The solutions are: pi/2, 3pi/2, 3pi/4, and 5pi/4.