Expand the expression using the sum & difference identities and simplify.sin(−θ−π2)\sin\left(-\theta-\frac{\pi}{2}\right)sin(−θ−2π)

− cos ⁡ θ -\cos\theta − cos θ

Expand the expression using the sum & difference identities and simplify. sin ( − θ − π 2 ) \sin\left(-\theta-\frac{\pi}{2}\right) sin ( − θ − 2 π )

Here we're asked to expand the expression using our sum and difference identities and then simplify. Now the expression that we're given here is the sine of negative theta minus pi over 2. So let's go ahead and use our difference formula for sine in order to expand this out. Now doing that will give me sine of negative theta times the cosine of pi over 2, and then subtracting the cosine of negative theta times the sine of pi over 2. Now the cosine of pi over 2 and the sine of pi over 2 are two values that I know. The cosine of pi over 2 is equal to 0 and the sine of pi over 2 is equal to 1. So here this entire term goes away because it's being multiplied by 0. Then I'm just left with a negative cosine of negative theta because that one multiplying this just leaves me with this. Now negative cosine of negative theta. We do know another identity that could help us simplify this further because the cosine of negative theta, knowing that cosine is an even function, is really just the cosine of theta. So all that I'm left with here is negative cosine of theta and this is my final answer. Thanks for watching and I'll see you in the next one.

Expand the expression using the sum & difference identities and simplify. sin ⁡ ( − θ − π 2 ) \sin\left(-\theta-\frac{\pi}{2}\right) sin ( − θ − 2 π )

− cos ⁡ θ -\cos\theta − cos θ

− sin ⁡ θ − cos ⁡ θ -\sin\theta-\cos\theta − sin θ − cos θ

− sin ⁡ θ -\sin\theta − sin θ

12. Trigonometric Identities

Sum and Difference Identities

Precalculus

12. Trigonometric Identities

Sum and Difference Identities

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Multiple Choice

Expand the expression using the sum & difference identities and simplify.
$\sin\left(-\theta-\frac{\pi}{2}\right)$

$-\sin\theta-\cos\theta$

$-\sin\theta$

$-\cos\theta$

Verified step by step guidance

Recognize that the expression involves the sine of a sum: $ \sin(-\theta - \frac{\pi}{2}) $. We can use the sine sum identity: $ \sin(a + b) = \sin a \cos b + \cos a \sin b $.

Rewrite the expression using the identity: $ \sin(-\theta - \frac{\pi}{2}) = \sin(-\theta) \cos(-\frac{\pi}{2}) + \cos(-\theta) \sin(-\frac{\pi}{2}) $.

Recall the values of trigonometric functions at specific angles: $ \cos(-\frac{\pi}{2}) = 0 $ and $ \sin(-\frac{\pi}{2}) = -1 $.

Substitute these values into the expression: $ \sin(-\theta) \cdot 0 + \cos(-\theta) \cdot (-1) $.

Simplify the expression: $ 0 - \cos(-\theta) $. Since $ \cos(-\theta) = \cos(\theta) $, the expression simplifies to $ -\cos(\theta) $.

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