Identities & Reciprocal Functions — practice questions

Simplify $\dfrac{1 - \cos^2\theta}{\cos\theta\tan\theta}$.

Prove that $\sin^4\theta - \cos^4\theta = 1 - 2\cos^2\theta$.

Express $3\sin x + 4\cos x$ in the form $R\sin(x + \alpha)$ where $R > 0$ and $0 < \alpha < \frac{\pi}{2}$. Hence state the maximum and minimum values of the expression and the values of $x$ at which they occur.

State the period, amplitude and range of $y = 3\sin(4\theta) + 2$.

Explain why the equation $2\csc\theta = 1$ has no solutions, and state the range of values of $k$ for which $\csc\theta = k$ does have solutions.

A Ferris wheel of radius 9 m has its centre 11 m above the ground and completes one revolution every 40 seconds. A rider boards at the lowest point at $t = 0$. Find a model for the rider's height, and determine for how long during each revolution the rider is more than 15 m above the ground.