Use the even-odd identities to evaluate the expression. − cot ( θ ) ⋅ sin ( − θ ) -\cot\left(\theta\right)\cdot\sin\left(-\theta\right) − cot ( θ ) ⋅ sin ( − θ )

Here we're asked to use our even odd identities in order to evaluate the expression, and the expression that we're given is the negative cotangent of theta times the sine of negative theta. Now here the first thing I'm going to do is rewrite the sine of negative theta to have a positive argument using my even odd identities. So the sine of negative theta can be replaced with negative sine of theta. So taking that and applying it here, this is negative sine of theta and this is getting multiplied by negative cotangent of theta. Now you might be tempted here to just stop because I just have 2 trig functions that are getting multiplied by each other so it's fine. But we can actually simplify this much further because we know that the cotangent of theta is simply equal to the cosine of theta divided by the sine of theta. Now make sure you keep that negative there. That's still there. It doesn't go away. And this is still getting multiplied by negative sine of theta. Now from here, I see that I have a sine in the top and in the bottom, so that's going to cancel out. Now I also have 2 negative signs, which means that I'm going to end up with a positive thing at the end. So all that I'm left with here is positive cosine of theta, and we're done. We have successfully simplified the negative cotangent of theta times the sine of negative negative theta all the way down to the cosine of theta. Thanks for watching and I'll see you in the next one.

Use the even-odd identities to evaluate the expression. − cot ⁡ ( θ ) ⋅ sin ⁡ ( − θ ) -\cot\left(\theta\right)\cdot\sin\left(-\theta\right) − cot ( θ ) ⋅ sin ( − θ )

− cos ⁡ θ -\cos\theta − cos θ

cos ⁡ θ sin ⁡ 2 θ \frac{\cos\theta}{\sin^2\theta} s i n 2 θ c o s θ

12. Trigonometric Identities

Introduction to Trigonometric Identities

Precalculus

12. Trigonometric Identities

Introduction to Trigonometric Identities

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Multiple Choice

Use the even-odd identities to evaluate the expression.
$-\cot\left(\theta\right)\cdot\sin\left(-\theta\right)$

$\tan\theta$

$-\cos\theta$

$\cos\theta$

$\frac{\cos\theta}{\sin^2\theta}$

Verified step by step guidance

Identify the even-odd identities for the trigonometric functions involved. The cotangent function, $ \cot(\theta) $, is an odd function, meaning $ \cot(-\theta) = -\cot(\theta) $. The sine function, $ \sin(\theta) $, is also an odd function, so $ \sin(-\theta) = -\sin(\theta) $.

Apply the even-odd identities to the expression $ -\cot(\theta) \cdot \sin(-\theta) $. Using the identities, this becomes $ -\cot(\theta) \cdot (-\sin(\theta)) = \cot(\theta) \cdot \sin(\theta) $.

Recognize that $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $. Therefore, the expression $ \tan(\theta) $ can be rewritten in terms of sine and cosine.

Substitute the expressions into the given equation and simplify. The expression becomes $ \frac{\cos(\theta)}{\sin^2(\theta)} $ after simplification.

Verify that the simplified expression matches the given correct answer, $ \frac{\cos(\theta)}{\sin^2(\theta)} $, confirming the use of even-odd identities and simplification steps were correctly applied.