## Exam-Style Question on Calculus## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 653. This question is similar to one that appeared on an IB AA Higher paper in 2023. The use of a calculator is allowed.

Consider the functions \(f(x)=\sin(x) \text{ and } g(x) = \tan(x+\frac{\pi}{2})\) in the region where \(\frac{\pi}{2}\le x \le \pi\)

The graphs \(y=f (x)\) and \(y = g(x)\) intersect at a point A whose x-coordinate is \(a\).

(a) Show that \(\sin^2{a}=-\cos{a}\).

(b) Hence, show that the tangents to the curves at A intersect at right angles.

Find the value of \(\cos(a)\). Give your answer in the form \( \dfrac{a + \sqrt{b}}{c} \).

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