Solve for t 4cos^2t-1=sin^2t+sint*cost

To solve for t means to find the angle t from the given
identity. We'll transform the given identity into a homogenous equation by substituting
1 by (sin t)^2 + (cos t)^2 = 1 and moving all terms to one
side.

(sin t)^2 + sint*cost -  4(cos t)^2 + (sin t)^2 +
(cos t)^2 = 0

We'll combine like
terms:

2(sin t)^2 + sint*cost - 3(cos t)^2 =
0

Since cos t is different from zero, we'll divide the
entire equation by (cos t)^2:

2(sin t)^2/(cos t)^2 +
sint*cost/(cos t)^2 - 3 = 0

According to the rule, the
ratio sin t/cos t = tan t.

2 (tan t)^2 + tan t - 3 =
0

We'll substitute tan t =
x:

2x^2 + x - 3 = 0

We'll
apply the quadratic formula:

x1 =
[-1+sqrt(1+24)]/4

x1 =
(-1+5)/4

x1 = 1

x2 =
(-1-5)/4

x2 = -3/2

We'll put
tan t = x1:

tan t = 1

t =
arctan 1 + k*pi

t = pi/4 +
k*pi

tan t =
x2

tan t =
-3/2

t = - arctan (3/2) +
k*pi

The solutions of the
equation are the values of t
angle:

{pi/4 + k*pi} U {-
arctan (3/2) + k*pi}