STEP II 1990 question 2 solution

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Using the product to sum and compound angle formulae:

This is because cos(x) is an even function.

For the second part, draw the cyclic quadrilateral with vertices P, Q, R, S, and also the diagonals PR and QS.

Let angle(PRQ) = angle(PSQ) = A, angle(QSR) = angle(QPR) = B, angle(RQS) = angle(RPS) = C, angle(SRP) = angle(SQP) = D. Each pair of angles is equal as they are angles subtended by the same arc.

It is clear that , as the angles make up the angles of a triangle.

This is a result of the (full) Sine Rule. R is the radius of the circle.

Using the result of the first part, you get:

Solution by Dystopia.