tan(180-x)sin(360-x)cos90-x)cos(-x)/sin(x-180)cos(450+x)tan(180-x)
To simplify the expression tan(180-x)sin(360-x)cos(90-x)cos(-x)/sin(x-180)cos(450+x)tan(180-x), we can apply trigonometric identities and simplify each term.
Let's start by simplifying tan(180-x):
Using the identity tan(180-x) = -tan(x), the expression becomes -tan(x).
Next, let's simplify sin(360-x):
Using the identity sin(360-x) = sin(x), the expression remains as sin(x).
Now, let's simplify cos(90-x):
Using the identity cos(90-x) = sin(x), the expression becomes sin(x).
For cos(-x), we know that cos(-x) = cos(x), so the term remains unchanged.
Next, let's simplify sin(x-180):
Using the identity sin(x-180) = -sin(x), the expression becomes -sin(x).
Now, let's simplify cos(450+x):
Using the identity cos(450+x) = -sin(x), the expression becomes -sin(x).
Finally, for tan(180-x), we already simplified it to -tan(x).
Now, let's substitute these simplified values back into the expression:
(-tan(x) * sin(x) * sin(x) * cos(x)) / (-sin(x) * cos(x) * -sin(x) * -tan(x))
Simplifying further:
(tan(x) * sin(x) * sin(x) * cos(x)) / (sin(x) * cos(x) * sin(x) * tan(x))
= (sin(x) * sin(x) * cos(x)) / (sin(x) * cos(x) * sin(x))
= cos(x) / 1
= cos(x)
Therefore, tan(180-x)sin(360-x)cos(90-x)cos(-x)/sin(x-180)cos(450+x)tan(180-x) simplifies to cos(x).