Question

Cos11a+cos3a/sin9a+sinaУпростить выражение

Answer 1

Не уверена,  но вроде бы так решается

Answer 2

To simplify the given expression (cos^11(a) + cos^3(a)) / (sin^9(a) + sin(a)), we can use trigonometric identities to rewrite the numerator and the denominator.

Let's start with the numerator: cos^11(a) + cos^3(a)

Using the identity: cos^2(a) = (1 + cos(2a)) / 2, we can rewrite cos^3(a) as follows: cos^3(a) = cos^2(a) * cos(a) = (1 + cos(2a)) / 2 * cos(a)

Now, we can rewrite the numerator: cos^11(a) + cos^3(a) = cos^10(a) * cos(a) + (1 + cos(2a)) / 2 * cos(a)

Next, let's look at the denominator: sin^9(a) + sin(a)

Using the identity: sin^2(a) = (1 - cos(2a)) / 2, we can rewrite sin^9(a) as follows: sin^9(a) = (sin^2(a))^4 = ((1 - cos(2a)) / 2)^4

Now, we can rewrite the denominator: sin^9(a) + sin(a) = ((1 - cos(2a)) / 2)^4 + sin(a)

Now, the expression becomes: (cos^10(a) * cos(a) + (1 + cos(2a)) / 2 * cos(a)) / (((1 - cos(2a)) / 2)^4 + sin(a))

To proceed further, we need to simplify the expression in the denominator: (((1 - cos(2a)) / 2)^4 + sin(a))

At this point, the expression is already simplified as much as possible. There are no further common identities to apply or any other straightforward simplifications that can be made. Therefore, the expression remains as it is:

(cos^10(a) * cos(a) + (1 + cos(2a)) / 2 * cos(a)) / (((1 - cos(2a)) / 2)^4 + sin(a))