Question

Cos^2(п-а)/tg^2(a-3п/2)+sin^2(2п-а)/tg^2(a-п/2)

Answer 1

Ответ:

$\frac{ { \cos}^{2} (\pi - \alpha )}{ {tg}^{2} ( \alpha - \frac{3\pi}{2} )} + \frac{ { \sin}^{2}(2\pi - \alpha ) }{ {tg}^{2} ( \alpha - \frac{\pi}{2}) } = \\ = \frac{ { \cos}^{2} ( \alpha )}{ {ctg}^{2} ( \alpha )} + \frac{ { \sin }^{2}( \alpha )}{ {ctg}^{2} ( \alpha )} = \frac{ { \cos}^{2}( \alpha ) + { \sin }^{2} ( \alpha )}{ {ctg}^{2}( \alpha ) } = \\ = \frac{1}{ {ctg}^{2} ( \alpha )} = {tg}^{2} ( \alpha )$

Answer 2

To simplify the expression:

cos^2(π - a) / tan^2(a - 3π/2) + sin^2(2π - a) / tan^2(a - π/2)

We can use some trigonometric identities and properties. Let's work step by step:

1. cos^2(π - a) = cos^2(a) (because cos(π - a) = -cos(a))

2. sin(2π - a) = -sin(a) (because sin(2π - a) = -sin(a))

3. tan(x) = sin(x) / cos(x)

Now, let's rewrite the expression using these identities:

cos^2(a) / (sin(a) / cos(a))^2 + (-sin(a))^2 / (sin(a) / cos(a))^2

Now, we'll simplify further:

1. cos^2(a) / (sin(a) / cos(a))^2 = (cos^2(a) * cos^2(a)) / sin^2(a) = cos^4(a) / sin^2(a)

2. (-sin(a))^2 / (sin(a) / cos(a))^2 = (sin^2(a) / sin^2(a)) * cos^2(a) = cos^2(a)

Now, we can put these simplified expressions back into the original expression:

(cos^4(a) / sin^2(a)) + cos^2(a)

To combine these terms, we need a common denominator:

Common denominator = sin^2(a)

So, we'll rewrite each term with the common denominator:

(cos^4(a) + cos^2(a) * sin^2(a)) / sin^2(a)

Now, we have a single fraction:

(cos^4(a) + cos^2(a) * sin^2(a)) / sin^2(a)

This is the simplified form of the given expression.