Answer:
See Below.
Step-by-step explanation:
We want to verify the equation:
![\displaystyle \frac{1}{1+\sin\theta} = \sec^2\theta - \sec\theta \tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B1%2B%5Csin%5Ctheta%7D%20%3D%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%20%5Ctan%5Ctheta)
To start, we can multiply the fraction by (1 - sin(θ)). This yields:
![\displaystyle \frac{1}{1+\sin\theta}\left(\frac{1-\sin\theta}{1-\sin\theta}\right) = \sec^2\theta - \sec\theta \tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B1%2B%5Csin%5Ctheta%7D%5Cleft%28%5Cfrac%7B1-%5Csin%5Ctheta%7D%7B1-%5Csin%5Ctheta%7D%5Cright%29%20%3D%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%20%5Ctan%5Ctheta)
Simplify. The denominator uses the difference of two squares pattern:
![\displaystyle \frac{1-\sin\theta}{\underbrace{1-\sin^2\theta}_{(a+b)(a-b)=a^2-b^2}} = \sec^2\theta - \sec\theta \tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1-%5Csin%5Ctheta%7D%7B%5Cunderbrace%7B1-%5Csin%5E2%5Ctheta%7D_%7B%28a%2Bb%29%28a-b%29%3Da%5E2-b%5E2%7D%7D%20%3D%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%20%5Ctan%5Ctheta)
Recall that sin²(θ) + cos²(θ) = 1. Hence, cos²(θ) = 1 - sin²(θ). Substitute:
![\displaystyle \displaystyle \frac{1-\sin\theta}{\cos^2\theta} = \sec^2\theta - \sec\theta \tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cdisplaystyle%20%5Cfrac%7B1-%5Csin%5Ctheta%7D%7B%5Ccos%5E2%5Ctheta%7D%20%3D%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%20%5Ctan%5Ctheta)
Split into two separate fractions:
![\displaystyle \frac{1}{\cos^2\theta} -\frac{\sin\theta}{\cos^2\theta} = \sec^2\theta - \sec\theta\tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B%5Ccos%5E2%5Ctheta%7D%20-%5Cfrac%7B%5Csin%5Ctheta%7D%7B%5Ccos%5E2%5Ctheta%7D%20%3D%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%5Ctan%5Ctheta)
Rewrite the two fractions:
![\displaystyle \left(\frac{1}{\cos\theta}\right)^2-\frac{\sin\theta}{\cos\theta}\cdot \frac{1}{\cos\theta}=\sec^2\theta - \sec\theta \tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28%5Cfrac%7B1%7D%7B%5Ccos%5Ctheta%7D%5Cright%29%5E2-%5Cfrac%7B%5Csin%5Ctheta%7D%7B%5Ccos%5Ctheta%7D%5Ccdot%20%5Cfrac%7B1%7D%7B%5Ccos%5Ctheta%7D%3D%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%20%5Ctan%5Ctheta)
By definition, 1 / cos(θ) = sec(θ) and sin(θ)/cos(θ) = tan(θ). Hence:
![\displaystyle \sec^2\theta - \sec\theta\tan\theta \stackrel{\checkmark}{=} \sec^2\theta - \sec\theta\tan\theta](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%5Ctan%5Ctheta%20%5Cstackrel%7B%5Ccheckmark%7D%7B%3D%7D%20%20%5Csec%5E2%5Ctheta%20-%20%5Csec%5Ctheta%5Ctan%5Ctheta)
Hence verified.
Answer:
8y=-11x-48
You have to subtract 11x bc the standard for is y=mx+b
Answer:
dont have one
Step-by-step explanation:
Given the equation:
![\frac{\sin^2x+\text{cos}^2x}{\cos x}=\sec x](https://tex.z-dn.net/?f=%5Cfrac%7B%5Csin%5E2x%2B%5Ctext%7Bcos%7D%5E2x%7D%7B%5Ccos%20x%7D%3D%5Csec%20x)
Let's determine the trigonometric identity that you could be used to verify the exquation.
Let's determine the identity:
Apply the trigonometric identity:
![\sin ^2x+\cos ^2x=1](https://tex.z-dn.net/?f=%5Csin%20%5E2x%2B%5Ccos%20%5E2x%3D1)
![\cos x=\frac{1}{\sec x}](https://tex.z-dn.net/?f=%5Ccos%20x%3D%5Cfrac%7B1%7D%7B%5Csec%20x%7D)
Replace cosx for 1/secx
Thus, we have:
![\begin{gathered} \frac{\sin^2x+\cos^2x}{\frac{1}{\sec x}} \\ \\ =(\sin ^2x+\cos ^2x)(\sec x) \\ \text{Where:} \\ (\sin ^2x+\cos ^2x)=1 \\ \\ We\text{ have:} \\ (\sin ^2x+\cos ^2x)(\sec x)=1\sec x=\sec x \end{gathered}](https://tex.z-dn.net/?f=%5Cbegin%7Bgathered%7D%20%5Cfrac%7B%5Csin%5E2x%2B%5Ccos%5E2x%7D%7B%5Cfrac%7B1%7D%7B%5Csec%20x%7D%7D%20%5C%5C%20%20%5C%5C%20%3D%28%5Csin%20%5E2x%2B%5Ccos%20%5E2x%29%28%5Csec%20x%29%20%5C%5C%20%5Ctext%7BWhere%3A%7D%20%5C%5C%20%28%5Csin%20%5E2x%2B%5Ccos%20%5E2x%29%3D1%20%5C%5C%20%20%5C%5C%20We%5Ctext%7B%20have%3A%7D%20%5C%5C%20%28%5Csin%20%5E2x%2B%5Ccos%20%5E2x%29%28%5Csec%20x%29%3D1%5Csec%20x%3D%5Csec%20x%20%5Cend%7Bgathered%7D)
The equation is an identity.
Therefore, the trignonometric identity you would use to verify the equation is:
![\cos ^2x+\sin ^2x=1](https://tex.z-dn.net/?f=%5Ccos%20%5E2x%2B%5Csin%20%5E2x%3D1)
ANSWER: