X
=
−
31
12
+
1
12
1249
or
x
=
−
31
12
+
−
1
12
1249
What is the t and what is the h
Add 8.
Go backwards on a number line from -4 8 times.
Power and chain rule (where the power rule kicks in because
):
![\left(\sqrt{\dfrac{\cos(2x)}{1+\sin(2x)}}\right)'=\dfrac1{2\sqrt{\frac{\cos(2x)}{1+\sin(2x)}}}\left(\dfrac{\cos(2x)}{1+\sin(2x)}\right)'](https://tex.z-dn.net/?f=%5Cleft%28%5Csqrt%7B%5Cdfrac%7B%5Ccos%282x%29%7D%7B1%2B%5Csin%282x%29%7D%7D%5Cright%29%27%3D%5Cdfrac1%7B2%5Csqrt%7B%5Cfrac%7B%5Ccos%282x%29%7D%7B1%2B%5Csin%282x%29%7D%7D%7D%5Cleft%28%5Cdfrac%7B%5Ccos%282x%29%7D%7B1%2B%5Csin%282x%29%7D%5Cright%29%27)
Simplify the leading term as
![\dfrac1{2\sqrt{\frac{\cos(2x)}{1+\sin(2x)}}}=\dfrac{\sqrt{1+\sin(2x)}}{2\sqrt{\cos(2x)}}](https://tex.z-dn.net/?f=%5Cdfrac1%7B2%5Csqrt%7B%5Cfrac%7B%5Ccos%282x%29%7D%7B1%2B%5Csin%282x%29%7D%7D%7D%3D%5Cdfrac%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D%7B2%5Csqrt%7B%5Ccos%282x%29%7D%7D)
Quotient rule:
![\left(\dfrac{\cos(2x)}{1+\sin(2x)}\right)'=\dfrac{(1+\sin(2x))(\cos(2x))'-\cos(2x)(1+\sin(2x))'}{(1+\sin(2x))^2}](https://tex.z-dn.net/?f=%5Cleft%28%5Cdfrac%7B%5Ccos%282x%29%7D%7B1%2B%5Csin%282x%29%7D%5Cright%29%27%3D%5Cdfrac%7B%281%2B%5Csin%282x%29%29%28%5Ccos%282x%29%29%27-%5Ccos%282x%29%281%2B%5Csin%282x%29%29%27%7D%7B%281%2B%5Csin%282x%29%29%5E2%7D)
Chain rule:
![(\cos(2x))'=-\sin(2x)(2x)'=-2\sin(2x)](https://tex.z-dn.net/?f=%28%5Ccos%282x%29%29%27%3D-%5Csin%282x%29%282x%29%27%3D-2%5Csin%282x%29)
![(1+\sin(2x))'=\cos(2x)(2x)'=2\cos(2x)](https://tex.z-dn.net/?f=%281%2B%5Csin%282x%29%29%27%3D%5Ccos%282x%29%282x%29%27%3D2%5Ccos%282x%29)
Put everything together and simplify:
![\dfrac{\sqrt{1+\sin(2x)}}{2\sqrt{\cos(2x)}}\dfrac{(1+\sin(2x))(-2\sin(2x))-\cos(2x)(2\cos(2x))}{(1+\sin(2x))^2}](https://tex.z-dn.net/?f=%5Cdfrac%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D%7B2%5Csqrt%7B%5Ccos%282x%29%7D%7D%5Cdfrac%7B%281%2B%5Csin%282x%29%29%28-2%5Csin%282x%29%29-%5Ccos%282x%29%282%5Ccos%282x%29%29%7D%7B%281%2B%5Csin%282x%29%29%5E2%7D)
![=\dfrac{\sqrt{1+\sin(2x)}}{2\sqrt{\cos(2x)}}\dfrac{-2\sin(2x)-2\sin^2(2x)-2\cos^2(2x)}{(1+\sin(2x))^2}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D%7B2%5Csqrt%7B%5Ccos%282x%29%7D%7D%5Cdfrac%7B-2%5Csin%282x%29-2%5Csin%5E2%282x%29-2%5Ccos%5E2%282x%29%7D%7B%281%2B%5Csin%282x%29%29%5E2%7D)
![=\dfrac{\sqrt{1+\sin(2x)}}{2\sqrt{\cos(2x)}}\dfrac{-2\sin(2x)-2}{(1+\sin(2x))^2}](https://tex.z-dn.net/?f=%3D%5Cdfrac%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D%7B2%5Csqrt%7B%5Ccos%282x%29%7D%7D%5Cdfrac%7B-2%5Csin%282x%29-2%7D%7B%281%2B%5Csin%282x%29%29%5E2%7D)
![=-\dfrac{\sqrt{1+\sin(2x)}}{\sqrt{\cos(2x)}}\dfrac{\sin(2x)+1}{(1+\sin(2x))^2}](https://tex.z-dn.net/?f=%3D-%5Cdfrac%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D%7B%5Csqrt%7B%5Ccos%282x%29%7D%7D%5Cdfrac%7B%5Csin%282x%29%2B1%7D%7B%281%2B%5Csin%282x%29%29%5E2%7D)
![=-\dfrac{\sqrt{1+\sin(2x)}}{\sqrt{\cos(2x)}}\dfrac1{1+\sin(2x)}](https://tex.z-dn.net/?f=%3D-%5Cdfrac%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D%7B%5Csqrt%7B%5Ccos%282x%29%7D%7D%5Cdfrac1%7B1%2B%5Csin%282x%29%7D)
![=-\dfrac1{\sqrt{\cos(2x)}}\dfrac1{\sqrt{1+\sin(2x)}}](https://tex.z-dn.net/?f=%3D-%5Cdfrac1%7B%5Csqrt%7B%5Ccos%282x%29%7D%7D%5Cdfrac1%7B%5Csqrt%7B1%2B%5Csin%282x%29%7D%7D)
![=\boxed{-\dfrac1{\sqrt{\cos(2x)(1+\sin(2x))}}}](https://tex.z-dn.net/?f=%3D%5Cboxed%7B-%5Cdfrac1%7B%5Csqrt%7B%5Ccos%282x%29%281%2B%5Csin%282x%29%29%7D%7D%7D)