Question

If cos alpha + sin alpha equal to root 2 cos alpha then prove that cos alpha minus sin alpha equal to root 2 sin alpha

Answer 1

Hope it may help you...

Answer 2

$We \:have \: cos \alpha+sin \alpha= \sqrt{2} cos \alpha \:--(1)$

/*On squaring both sides, we get */

$\implies (cos \alpha+sin \alpha )^{2}= (\sqrt{2} cos \alpha)^{2}$

$\implies cos^{2}\alpha + sin^{2} \alpha + 2sin \alpha cos \alpha = 2cos^{2} \alpha$

$\implies sin^{2} \alpha = 2cos^{2}\alpha - cos^{2} \alpha - 2sin \alpha cos \alpha$

$\implies sin^{2} \alpha = cos^{2}\alpha - 2sin \alpha cos \alpha$

$\implies sin^{2} \alpha + sin^{2} \alpha = cos^{2}\alpha+ sin^{2} \alpha - 2sin \alpha cos \alpha$

$\implies 2sin^{2} \alpha = ( cos \alpha - sin \alpha )^{2}$

$\implies (\sqrt{2}sin \alpha )^{2} = ( cos \alpha - sin \alpha )^{2}$

$\implies \sqrt{2} sin \alpha = cos\alpha - sin \alpha$

$\implies cos\alpha - sin \alpha = \sqrt{2} sin \alpha$

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