Question

Prove that:- cos(π+x) = - cos x​

Answer 1

Step-by-step explanation:

Triangle OAB ~ triangle ODC ( by AA similarity)

=> OA/OD = OB/OC ( csst)

Or, OA/OB = OD/OC . . . . . . . .( 1)

When initial ray is rotated (180° + x)° , it takes position OB.

So in tri OAB, OA is negative.

=> cos ( pi + x) = - ve OA/OB ( as, angle is in 3rd quadrant ) . . . . .(2)

& In triangle ODC,

cos x = OD/OC

Or, cos x = OA/OB ( by eq 1)

By putting this value of OA/OB in( eq 2)

cos ( pi + x) = - cos x

Answer 2

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$cos(\pi + x)$

$= cos\pi \: \times cosx - sin\pi \times sinx$

$= - 1 \times \: cos x - 0 \times sinx$

$= - cosx$

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