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 argument using my even odd identities. So the sign of negative theta can be replaced with negative sign of theta. So taking that and applying it here, this is negative sign 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 two 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 two 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 Functions

Trigonometric Identities

Business Calculus

12. Trigonometric Functions

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

Step 1: Recall the even-odd identities for trigonometric functions. The even-odd identities state that: sin(-θ) = -sin(θ), cos(-θ) = cos(θ), and cot(-θ) = -cot(θ). These identities will help simplify the given expression.

Step 2: Simplify the term -cot(θ)⋅sin(-θ). Using the even-odd identities, replace sin(-θ) with -sin(θ). This makes the term -cot(θ)⋅(-sin(θ)), which simplifies to cot(θ)⋅sin(θ).

Step 3: Analyze the denominator tan(θ) - cos(θ). Recall that tan(θ) = sin(θ)/cos(θ). Substitute this into the denominator to rewrite it as (sin(θ)/cos(θ)) - cos(θ). Combine the terms over a common denominator, which is cos(θ).

Step 4: Combine the simplified numerator and denominator. The numerator is cot(θ)⋅sin(θ), and the denominator is the simplified expression from Step 3. Use the definition of cot(θ) = cos(θ)/sin(θ) to further simplify the numerator.

Step 5: Simplify the entire expression. After substituting and simplifying, the final result should match the given correct answer, cos(θ)/sin²(θ). Verify that all steps align with the correct answer.