Which expression is equivalent to tan(-x)cos^2x?
-sin xcos x
sin xcos x
tan x + sin2x
tan x − sin2x
-tan x − sin2x
tan(-x)=sin(-x)/cosx
then tan(-x)cos^2x=sin(-x)cosx=-sinxcosx
To find the expression that is equivalent to tan(-x)cos^2x, we can use trigonometric identities and basic properties of trigonometric functions.
First, let's recall some identities:
1. tan(-x) = -tan(x) (tangent function is an odd function)
2. cos^2(x) = (cos(x))^2
Now, substitute these identities into the given expression:
tan(-x)cos^2x = -tan(x)(cos(x))^2
Since there are no options that exactly match this expression, let's simplify it further:
- tan(x)(cos(x))^2 = -sin(x)/cos(x)(cos(x))^2
Using the identity sin(x)/cos(x) = tan(x):
- sin(x)/cos(x)(cos(x))^2 = -sin(x)/cos^2(x)
Now, simplify further using the quotient identity for tangent:
- sin(x)/cos^2(x) = -tan(x)
Therefore, we can see that the expression -tan(x) is equivalent to tan(-x)cos^2x.
So, the correct option is: -tan x.
To find the expression that is equivalent to tan(-x)cos^2x, we can use trigonometric identities.
One identity that might be useful is: tan(-x) = -tan(x).
Using this identity, we can rewrite the expression as: -tan(x)cos^2x.
Another identity that might be useful is: cos^2x = 1 - sin^2x.
Using this identity, we can further rewrite the expression as: -tan(x)(1 - sin^2x).
Expanding this expression, we get: -tan(x) + tan(x)sin^2x.
Therefore, the expression that is equivalent to tan(-x)cos^2x is -tan(x) + tan(x)sin^2x.
So, the correct answer is "tan x − sin2x".