Open Challenge :)

Let x,yRx,y\quad \in \quad R such that cosxcosy+2siny+2sinxcosy=3\cos { x } \cos { y } \quad +\quad \quad 2\sin { y } \quad +\quad 2\sin { x } \cos { y } \quad =\quad 3.
Then find the value of : tan2x+5tan2y=?\tan ^{ 2 }{ x } +\quad 5\tan ^{ 2 }{ y } \quad =\quad ?.

It is really very beautiful question. That's why I share this with our Brilliant community.

Use any Tool of mathematics. There are no restrictions

(You may use Vectors, Complex numbes, Trigonometry, etc .)

Note by Deepanshu Gupta
4 years, 11 months ago

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Hint: Write the given equation as

(cosx+2sinx)cosy+2siny=3\bigg(\cos x+2\sin x\bigg)\cos y+2\sin y=3

¨\ddot\smile

Karthik Kannan - 4 years, 11 months ago

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@Karthik Kannan Did you mean this way :

E=(cosx+2sinx)cosy+2sinyEmax=(cosx+2sinx)2+(2)2=13232cos2x+2sin2xEmax[2,3]Emax3ButEmax=3E\quad \quad \quad =\quad (\cos x+2\sin x)\cos y+2\sin y\\ \\ { E }_{ max }\quad =\quad \sqrt { { (\cos x+2\sin x) }^{ 2 }+\quad { (2) }^{ 2 } } \quad \\ \\ \quad \quad \quad \quad =\quad \sqrt { \cfrac { 13 }{ 2 } \quad -\quad \cfrac { 3 }{ 2 } \cos { 2x } \quad +\quad 2\sin { 2x } } \\ \quad \quad \quad \quad \\ \quad { E }_{ max }\quad \in \quad \left[ 2\quad ,\quad 3 \right] \\ \\ { E }_{ max }\quad \le \quad 3\\ \\ But\quad \quad \quad { E }_{ max }\quad =\quad 3\\ .

Now using boundedness of function ! Great ! This is also Nice Method !!

Deepanshu Gupta - 4 years, 11 months ago

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if x,y belongs to R , then the first expression should be true for all real values of x, but we can see here its true only for some specific

sandeep Rathod - 4 years, 11 months ago

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Please answer my doubt deepanshu gupta

sandeep Rathod - 4 years, 11 months ago

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wait be clear , I said : x,yRx,y\quad \in \quad R. which means x and y are real numbers. it dosn't mean that that expression is true for every x,y .

And if i say x,yR\forall \quad x,y\quad \in \quad R\quad \quad . then it means this expression is true for every x ( which means it is identity )

Deepanshu Gupta - 4 years, 11 months ago

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HINT: USE cauchy inequality

Lakshya Kumar - 4 years, 11 months ago

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Yes that is best!!

Deepanshu Gupta - 4 years, 11 months ago

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could you please explain how?

Aman Gautam - 4 years, 11 months ago

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@Aman Gautam Let two Vectors such that :

\quad \overset { \rightarrow }{ A } \quad =\quad \cos { x } \cos { y } \quad +\quad \sin { y } +\quad \quad \sin { x } \cos { y } \\ \\ \quad \overset { \rightarrow }{ B } \quad =\quad \overset { \^ }{ i } \quad +\quad 2\overset { \^ }{ j } \quad +\quad 2\overset { \^ }{ k } \quad .

Now Verify that :

A.B=AB\overset { \rightarrow }{ A } .\overset { \rightarrow }{ B } \quad =\quad \left| \overset { \rightarrow }{ A } \right| \left| \overset { \rightarrow }{ B } \right| .

So angle between these two vectors is zero degree , means they are parallel ,

Now use simple condition of Parallel Vectors and get the Answer ! :)

@Aman Gautam I Hope you Got it !

Deepanshu Gupta - 4 years, 11 months ago

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