Question

If cos theta -sin theta = √2 sin theta, then prove that cos theta + sin theta = √2 cos theta


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Answer 1

Required Answer:-

Given:

• cos(x) - sin(x) = √2 sin(x)

To prove:

• cos(x) + sin(x) = √2 cos(x)

Proof:

$\rm \implies \cos(x) - \sin(x) = \sqrt{2} \sin(x)$

$\rm \implies \cos(x) = \sqrt{2} \sin(x) + \sin(x)$

$\rm \implies \cos(x) =\sin(x) \{ \sqrt{2} + 1 \}$

$\rm \implies \sin(x) = \dfrac{ \cos(x) }{( \sqrt{2} + 1)}$

$\rm \implies \sin(x) = \dfrac{ \cos(x)( \sqrt{2} - 1) }{( \sqrt{2} + 1)( \sqrt{2} - 1)}$

$\rm \implies \sin(x) = \dfrac{ \cos(x)( \sqrt{2} - 1) }{ {( \sqrt{2} )}^{2} - {(1)}^{2} }$

$\rm \implies \sin(x) = \dfrac{ \cos(x)( \sqrt{2} - 1) }{2 - 1}$

$\rm \implies \sin(x) = \cos(x)( \sqrt{2} - 1)$

$\rm \implies \sin(x) = \sqrt{2} \cos(x) - \cos(x)$

$\rm \implies \sin(x) + \cos(x) = \sqrt{2} \cos(x)$

(Hence Proved)

Relationship Between Trigo Functions:

• sin(x) = 1/cosec(x)
• cos(x) = 1/sec(x)
• tan(x) = 1/cot(x)
• sin(x)/cos(x) = tan(x)

Answer 2

Answer:

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