a)

2sin²x = 1 + cosx

=> 2( 1 - cos²x) = 1 + cosx

=> 2 - 2cos²x = 1 + cosx

=> 2cos²x + cosx - 1 = 0

=> (2 cos x - 1)(cos x + 1) = 0

cos x = -1 and 1/2

x = π/3, π and 5π/3 in the interval [0, 2π ]

b)

2tan²x = secx - 1

=> 2( sec²x - 1) = secx - 1

=> 2sec²x - 2 = secx - 1

=> 2sec²x - secx - 1 = 0

=> (2 sec x + 1) (sec x - 1) = 0

sec x = -1/2 and 1

only valid solution is sec x = 1

x = 0 and 2π

c)

(sinx)(cosx) = (cosx)

=> cos x ( sin x - 1) = 0

cos x = 0 and sin x = 1

x = π/2, 3π/2 and π/2

a) 2 sin² (x) = 1 + cos (x)

=> 2 ( 1 - cos² (x) ) = 1 + cos (x)

=> 2 - 2 cos² (x) = 1 + cos (x)

=> 1 - 2 cos² (x) - cos (x) = 0

=> -2 cos² (x) - cos (x) + 1 = 0

Let y = cos (x)

Then,

-2 cos²(x) - cos (x) + 1 = 0

becomes

-2y² - y + 1 = 0

-2y² - 2y + y + 1 = 0

(-2y² - 2y) (y+ 1) = 0

-2y (y + 1) (y + 1) = 0

( -2y+ 1) (y + 1) = 0

Now, substituting in cos (x) for y,

( -2y + 1) (y + 1) = 0

becomes

( -2 cos (x) + 1) ( cos (x) + 1) = 0

set each factor equal to 0 and solve for x:

-2 cos (x) + 1 = 0 <-- let's call this equation (1)

cos (x) + 1 = 0 <-- let's call this equation (2)

From equation (1),

cos (x) = ½

From the unit circle,

x = {60 + 360n , 300 + 360n} where n is any integer. I need to specify that because you didn't give me the restrictions for x.

From equation (2),

cos (x) = -1

From the unit circle again,

x = {180 + 360n} where n, once again, is any integer

You can solve b) and c) in similar manner.

Hope this helps!!

a) Use sin^2 + cos^2 =1 -> sin^2 = 1-cos^2

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

2 - 2 cos^2 = 1 + cos

2 cos^2 + cos - 1 = 0

(2cos - 1)( cos + 1 = 0)

cos x = 1/2 or cos x = -1

So x = 30 or 180

b) tan = sin/cos sec = 1/cos

2 (sin/cos)^2 = 1/cos - 1

Multiply by cos^2

2 sin^2 = cos - cos^2

Using sin^2 = 1-cos^2

2 - 2cos^2 = cos - cos^2

Cos^2 + cos - 2 = 0

(cos + 2)(cos - 1) = 0

cos x = 1 so x = 0 or 180

c) Cancel cos leaving sin x = 1 x = 90

Cos x = 0 -> x = 90 or 270

Learn or learn to derive your identities.

http://en.wikipedia.org/wiki/List_of_trigonometric...

http://en.wikipedia.org/wiki/List_of_trigonometric...