Math, asked by harshalbakde9, 10 days ago

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


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Answers

Answered by anindyaadhikari13
5

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)
Answered by Anisha5119
4

Answer:

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