Question

• Нужно упростить выражение: sin(90-a)*ctg(270-a) / sin(360-a)*ctg(180+a).​

Answer 1

$\frac{ \sin(90 {}^{\circ }- \alpha)\cdot \cot(270 {}^{\circ} - \alpha ) }{ \sin(360 {}^{\circ } - \alpha ) \cdot \cot(180 {}^{\circ} + \alpha ) } = \frac{ \cos \alpha\cdot \tan \alpha }{ - \sin \alpha \cdot \cot \alpha } = \frac{ \cos \alpha\cdot \frac{ \sin \alpha }{ \cos \alpha} }{ - \sin \alpha\cdot \frac{ \cos \alpha }{ \sin \alpha } } = - \frac{ \sin \alpha }{ \cos \alpha }$

Использовал формулу приведения и основного тригонометрического тождества .

Answer 2

Simplifying the Expression

To simplify the given expression sin(90-a)*ctg(270-a) / sin(360-a)*ctg(180+a), we can use trigonometric identities and properties to rewrite and simplify the expression.

First, let's rewrite the trigonometric functions using their respective identities: - sin(90-a) can be rewritten as cos(a) using the complementary angle identity for sine. - ctg(270-a) can be rewritten as tan(a) using the cotangent-tangent relationship. - sin(360-a) can be rewritten as -sin(a) using the periodicity of sine. - ctg(180+a) can be rewritten as -tan(a) using the cotangent-tangent relationship.

Now, let's substitute these rewritten expressions back into the original expression: cos(a)*tan(a) / (-sin(a))*(-tan(a))

Simplify further: cos(a)*tan(a) / sin(a)*tan(a)

Now, we can simplify by canceling out the common factor tan(a): cos(a) / sin(a)

Using the fundamental trigonometric identity tan(a) = sin(a) / cos(a), we can rewrite the expression as: cos(a) / sin(a) = 1 / tan(a)

So, the simplified expression is 1 / tan(a).

Therefore, the simplified form of the given expression sin(90-a)*ctg(270-a) / sin(360-a)*ctg(180+a) is 1 / tan(a).

Conclusion

The simplified form of the given expression is 1 / tan(a).