First slide

Introduction to ITF

Question

If $\sin^{- 1} x = θ + β$ and $\sin^{- 1} y = θ - β,$ then $1 + x y =$

Easy

A

$\sin^{2} θ + \sin^{2} β$
B

$\sin^{2} θ + \cos^{2} β$
C

$\cos^{2} θ + \cos^{2} θ$
D

$\cos^{2} θ + \sin^{2} β$

Solution

We have $x = \sin (θ + β)$ , and $y = \sin (θ - β)$
$\begin{array}{l} \Rightarrow 1 + x y = 1 + \sin (θ + β) \sin (θ - β) \\ = 1 + \sin^{2} θ - \sin^{2} β = \sin^{2} θ + \cos^{2} β \end{array}$

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