Explain why $\frac{PF}{EF} \times \frac{EF}{OF}$ in Line 4 leads to $\cos A \sin B$ in Line 5 - AQA - A-Level Maths Pure - Question 14 - 2018 - Paper 1

This expression can be simplified by recognizing the relationships between the sides of the triangles involved. In the right triangle OEF, we know that:

• $PF = EF \cos A$, which represents the adjacent side over the hypotenuse for angle A.
• $EF = OF \sin B$, which represents the opposite side over the hypotenuse for angle B.

Therefore, when we substitute these into the expression, we have:

$\frac{PF}{EF} = \frac{EF \cos A}{EF} = \cos A$

And since $\frac{EF}{OF} = \sin B$, it then becomes clear that the overall product leads to:

$\cos A \sin B$

This confirms the correct reasoning from Line 4 to Line 5.