Prove that \(\cos x + \sin x \tan x \equiv \frac { 1 } { \cos x }\) (where \(x \neq \frac { 1 } { 2 } n \pi\) for any odd integer \(n\) ).
Solve the equation \(2 \sin ^ { 2 } x = \cos ^ { 2 } x\) for \(0 ^ { \circ } \leqslant x \leqslant 180 ^ { \circ }\).
Show that \(B\) lies on \(A C\).
Find the ratio \(A B : B C\).
The diagram shows the line \(x + y = 6\) and the point \(P ( 2,4 )\) that lies on the line.
A copy of the diagram is given in the Printed Answer Booklet.
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The distinct point \(Q\) also lies on the line \(x + y = 6\) and is such that \(| \overrightarrow { O Q } | = | \overrightarrow { O P } |\), where \(O\) is the origin.
Find the magnitude and direction of the vector \(\overrightarrow { P Q }\).
The point \(R\) is such that \(\overrightarrow { P R }\) is perpendicular to \(\overrightarrow { O P }\) and \(| \overrightarrow { P R } | = \frac { 1 } { 2 } | \overrightarrow { O P } |\).
Write down the position vectors of the two possible positions of the point \(R\).