Wiskunde Multiple Choice Vraag uit Gricha's Wiskundige Vragenbank

tan x + tan( 90° + x) + tan( 180° + x) + tan( 270° + x) is gelijk aan _{( x is geen veelvoud van 90°)}	A. 0
	B. cot 2x
	C. − cot 2x
	D. − 4cot 2x
	E. 4cot 2x

[ 5-6857 - op net sinds 13.7.2020-(E)-8.11.2023 ]

Translation in E N G L I S H

tan x + tan( 90° + x) + tan( 180° + x) + tan( 270° + x) is equal to	A. 0
	B. cot 2x
	C. − (cot 2x)
	D. − 4(cot 2x)
	E. 4(cot 2x)

Oplossing - Solution

$\\{\color{Green}.}\;\;\tan x+\tan(90^\circ+x)+\tan(180^\circ+x)+\tan(270^\circ+x)\\=\tan x+\tan(90^\circ+x)+\tan x\qquad\qquad+\tan(90^\circ+x)\\=2.\tan x+2.\cot[90^\circ-(90^\circ+x)]\\=2.\tan x+2.\cot(-x)\\=2.\tan x-2.\cot x\\=2\left(\frac{\sin x}{\cos x}-\frac{\cos x}{\sin x}\right)=2.\frac{\sin^2x-\cos^2x}{\sin x.\cos x}=4.\frac{-\cos 2x}{\sin 2x}=-4\,\cot 2x$