Prove the identity \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 1 + \cos x } \equiv - \cos x\).
Hence solve the equation \(\frac { \sin ^ { 2 } x - \cos x - 1 } { 2 + 2 \cos x } = \frac { 1 } { 4 }\) for \(0 ^ { \circ } \leqslant x \leqslant 360 ^ { \circ }\).
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The function f is defined by \(\mathrm { f } ( x ) = \frac { 2 } { x ^ { 2 } } + 4\) for \(x < 0\). The diagram shows the graph of \(\mathrm { y } = \mathrm { f } ( \mathrm { x } )\).
On this diagram, sketch the graph of \(y = f ^ { - 1 } ( x )\). Show any relevant mirror line.
Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\).
Solve the equation \(\mathrm { f } ( x ) = 4.5\).
Explain why the equation \(\mathrm { f } ^ { - 1 } ( x ) = \mathrm { f } ( x )\) has no solution.
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In the diagram, \(A O D\) and \(B C\) are two parallel straight lines. Arc \(A B\) is part of a circle with centre \(O\) and radius 15 cm . Angle \(B O A = \theta\) radians. Arc \(C D\) is part of a circle with centre \(O\) and radius 10 cm . Angle \(C O D = \frac { 1 } { 2 } \pi\) radians.
Show that \(\theta = 0.7297\), correct to 4 decimal places.
Find the perimeter and the area of the shape \(A B C D\). Give your answers correct to 3 significant figures.