Use identities to find an equivalent expression in terms of sines and cosines only.
cot θ sin θ - tan θ cos θ
tan theta = sin theta / cos theta
cot theta = cos theta / sin theta
cot heta * sin theta - tan theta * cos theta =
( cos theta / sin theta ) * sin theta - ( sin theta / cos theta ) * cos theta =
cos theta - sin thets
(cos/sin)sin - (sin/cos)cos
= cos - sin
To find an equivalent expression in terms of sines and cosines only, we can use the trigonometric identities.
First, let's rewrite the expression using the reciprocal identities:
cot(θ) = 1 / tan(θ)
and
tan(θ) = sin(θ) / cos(θ)
Substituting these into the expression, we have:
(1 / tan(θ)) * sin(θ) - (sin(θ) / cos(θ)) * cos(θ)
Next, simplify further:
sin(θ) / tan(θ) - sin(θ)
To express this in terms of sines and cosines only, we need to find a common denominator for the fraction. The least common denominator is cos(θ), so we'll multiply both the numerator and the denominator of the first fraction by cos(θ):
(sin(θ) * cos(θ)) / (tan(θ) * cos(θ)) - sin(θ)
Using the identity sin(θ) * cos(θ) = (1/2) * sin(2θ) and tan(θ) = sin(θ) / cos(θ):
((1/2) * sin(2θ)) / (sin(θ) / cos(θ)) - sin(θ)
Now, we can simplify the expression further:
(1/2) * sin(2θ) * (cos(θ) / sin(θ)) - sin(θ)
The sin(θ) in the numerator cancels out with one of the sin(θ) in the denominator. We are left with:
(1/2) * sin(2θ) * cos(θ) - sin(θ)
Finally, we can write the equivalent expression in terms of sines and cosines only:
(1/2) * sin(2θ) * cos(θ) - sin(θ)
And that's the final answer!