Studdy AI Logo
Math  /  Trigonometry

Question2) semplifica l'espressione cos⁡(π−α)sin⁡(3π2+α)sin⁡2αtan⁡(α−π)+tan⁡(π2+α)\frac{\cos (\pi-\alpha) \sin \left(\frac{3 \pi}{2}+\alpha\right)}{\sin ^{2} \alpha} \tan (\alpha-\pi)+\tan \left(\frac{\pi}{2}+\alpha\right)

Studdy Solution

STEP 1

1. We are asked to simplify the given trigonometric expression.
2. We will use trigonometric identities to simplify the expression.

STEP 2

1. Simplify each trigonometric function using identities.
2. Simplify the entire expression by combining terms.

STEP 3

Simplify cos⁡(π−α)\cos(\pi - \alpha) using the identity cos⁡(π−α)=−cos⁡(α)\cos(\pi - \alpha) = -\cos(\alpha).
cos⁡(π−α)=−cos⁡(α) \cos(\pi - \alpha) = -\cos(\alpha)

STEP 4

Simplify sin⁡(3π2+α)\sin\left(\frac{3\pi}{2} + \alpha\right) using the identity sin⁡(3π2+α)=−cos⁡(α)\sin\left(\frac{3\pi}{2} + \alpha\right) = -\cos(\alpha).
sin⁡(3π2+α)=−cos⁡(α) \sin\left(\frac{3\pi}{2} + \alpha\right) = -\cos(\alpha)

STEP 5

Simplify tan⁡(α−π)\tan(\alpha - \pi) using the identity tan⁡(α−π)=tan⁡(α)\tan(\alpha - \pi) = \tan(\alpha).
tan⁡(α−π)=tan⁡(α) \tan(\alpha - \pi) = \tan(\alpha)

STEP 6

Simplify tan⁡(π2+α)\tan\left(\frac{\pi}{2} + \alpha\right) using the identity tan⁡(π2+α)=−cot⁡(α)\tan\left(\frac{\pi}{2} + \alpha\right) = -\cot(\alpha).
tan⁡(π2+α)=−cot⁡(α) \tan\left(\frac{\pi}{2} + \alpha\right) = -\cot(\alpha)

STEP 7

Substitute the simplified expressions back into the original expression:
cos⁡(π−α)sin⁡(3π2+α)sin⁡2αtan⁡(α−π)+tan⁡(π2+α)\frac{\cos (\pi-\alpha) \sin \left(\frac{3 \pi}{2}+\alpha\right)}{\sin ^{2} \alpha} \tan (\alpha-\pi) + \tan \left(\frac{\pi}{2}+\alpha\right)
becomes:
(−cos⁡(α))(−cos⁡(α))sin⁡2(α)tan⁡(α)−cot⁡(α)\frac{(-\cos(\alpha))(-\cos(\alpha))}{\sin^2(\alpha)} \tan(\alpha) - \cot(\alpha)

STEP 8

Simplify the first term:
cos⁡2(α)sin⁡2(α)tan⁡(α)=cot⁡2(α)tan⁡(α)\frac{\cos^2(\alpha)}{\sin^2(\alpha)} \tan(\alpha) = \cot^2(\alpha) \tan(\alpha)
Since cot⁡(α)=1tan⁡(α)\cot(\alpha) = \frac{1}{\tan(\alpha)}, we have:
cot⁡2(α)tan⁡(α)=1tan⁡(α)tan⁡(α)=cot⁡(α)\cot^2(\alpha) \tan(\alpha) = \frac{1}{\tan(\alpha)} \tan(\alpha) = \cot(\alpha)

STEP 9

Combine the terms:
cot⁡(α)−cot⁡(α)=0\cot(\alpha) - \cot(\alpha) = 0
The simplified expression is:
0 \boxed{0}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord