﻿ Simplify the trigonometric expression \dfrac{\sqrt{1-2\sin 40\degree \cos 40\degree } }{\cos 40\degree -\sqrt{1-\cos^2 40\degree } }

#### Question

Simplify the trigonometric expression

\dfrac{\sqrt{1-2\sin 40\degree \cos 40\degree } }{\cos 40\degree -\sqrt{1-\cos^2 40\degree } }

Collected in the board: Trigonometry

Steven Zheng posted 2 months ago

Using Pythagorean Identity to transform the radicand to the form of perfect square.

\dfrac{\sqrt{1-2\sin 40\degree \cos 40\degree } }{\cos 40\degree -\sqrt{1-\cos^2 40\degree } }

=\dfrac{\sqrt{\sin ^2 40\degree +\cos^2 40\degree -2\sin 40\degree \cos 40\degree } }{\cos 40\degree -\sqrt{1-\cos^2 40\degree } }

=\dfrac{\sqrt{(\cos 40\degree -\sin 40\degree )^2 } }{\cos 40\degree -\sqrt{\sin^2 40\degree } }

=\dfrac{|\cos 40\degree -\sin 40\degree |}{\cos 40\degree -|\sin 40\degree| }

\because \cos 40\degree >\sin 40\degree

\therefore \dfrac{|\cos 40\degree -\sin 40\degree |}{\cos 40\degree -|\sin 40\degree| }=1

Steven Zheng posted 2 months ago

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