Question

1) cosa/sin2a 2) sin2x/sinx - coax 3) cosAsin2a/tgA 4) cos2x-cos^2x

Answer 1

1. $\frac{cosa}{sin2a} =\frac{cosa}{2sinacosa} =\frac{1}{2sina}$
2. $\frac{sin2x}{sinx} -cosx = \frac{2sinxcosx}{sinx} -cosx = 2cosx-cosx=cosx$
3. $\frac{cosasin2a}{tga} =\frac{2sinacos^{2}a }{\frac{sina}{cosa} } =2cosa$
4. $cos2x-cos^{2}x =cos^{2} x-sin^{2} x-cos^{2} x=-sin^{2} x$

Answer 2

Let's simplify each expression step-by-step:

1. cosa/sin2a: We can simplify this expression using trigonometric identities. Recall that sin(2a) = 2 * sin(a) * cos(a). Therefore, sin^2(a) = (sin(a))^2.

cosa/sin2a = cosa / (2 * sin(a) * cos(a))

Now, we can cancel out common factors:

cosa / (2 * sin(a) * cos(a)) = 1 / (2 * sin(a))

So the simplified expression is 1 / (2 * sin(a)).

2. sin2x/sinx - coax: It seems there is a typo in the expression "coax." Assuming it was meant to be "cosx," let's proceed with the simplification.

sin2x/sinx - cosx: We can use the identity sin(2x) = 2 * sin(x) * cos(x) to simplify.

sin2x/sinx - cosx = (2 * sin(x) * cos(x)) / sin(x) - cos(x)

Now, we can cancel out the common factor sin(x):

(2 * sin(x) * cos(x)) / sin(x) - cos(x) = 2 * cos(x) - cos(x) = cos(x)

So the simplified expression is cos(x).

3. cosAsin2a/tgA: Let's use trigonometric identities to simplify this expression.

Recall that tan(A) = sin(A) / cos(A) and sin(2a) = 2 * sin(a) * cos(a).

cosAsin2a/tgA = (cos(A) * 2 * sin(a) * cos(a)) / (sin(A) / cos(A))

Now, we can simplify by canceling out common factors:

(cos(A) * 2 * sin(a) * cos(a)) / (sin(A) / cos(A)) = 2 * cos^2(a)

So the simplified expression is 2 * cos^2(a).

4. cos2x-cos^2x: We can use trigonometric identities to simplify this expression.

Recall that cos(2x) = 1 - 2 * sin^2(x).

cos2x - cos^2x = (1 - 2 * sin^2(x)) - cos^2(x)

Now, let's combine like terms:

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

Since sin^2(x) + cos^2(x) = 1 (from the Pythagorean identity), we can replace sin^2(x) in the expression:

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

Now distribute the 2:

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

Combine like terms:

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

So the simplified expression is -1 + cos^2(x).