Use an identity to solve the following equation on the interval \( [0,2 \pi) \). \( 2 \sin ^{2} x-\cos x-1=0 \) Select the correct choice below and, if necessary, fill in the answer box to com A. \( x= \) (Type your answer in radians. Use integers or fractions for any numbe needed. Use a comma to separate answers as needed.) B. There is no solution.

\( \frac{1}{1-\sin (\alpha )}+\frac{1}{1+\sin (\alpha )}-2\tan (^{2}\alpha )^{} \)

Aufgabe 4 (4 Punkte, zum präzisen Aufschrieb). Zeigen Sie, dass für alle \( x, y \in \mathbb{R} \) gilt: (a) \( \cos (2 x)=2 \cos ^{2} x-1 \) und \( \sin (2 x)=2 \sin x \cos x \). (b) \( \cos x-\cos y=2 \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{y-x}{2}\right) \) und \( \sin x-\sin y=2 \cos \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right) \).

a) Решите неравенство: \( \sin y<-\frac{1}{\sqrt{2}} \). b) Используя результаты предыдущего действия решите: \( \sin \left(3 x+\frac{\pi}{6}\right)<-\frac{\sqrt{2}}{2} \). c) Найдите решение неравенства: \( \frac{1}{2} \cos 3 x+\frac{\sqrt{3}}{2} \sin 3 x<-\frac{\sqrt{2}}{2} \).

6. A camera is filming the progress as a daredevil attempt to scale the wall of a skyscraper. The climber i moving vertically at a constant rate of 15 meters per minute, and the camera is 40 meters from the base of the skyscraper. Through how many radians per minute is the camera angle changing when the climbe is 30 meters feet up the building?

\( \frac{1}{1-\sin (\alpha )}+\frac{1}{1+\sin (\alpha )}-2\tan (^{2}\alpha )^{} \)