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  • Solve! - If ;, thenis equal to : - Trigonometry - JEE Main

 

If   \sin ^{4 }\alpha+4\cos ^{4}\beta+2=4\sqrt{2}\sin \alpha \cos \beta;  \alpha ,\beta \in \left [ 0,\pi \right ], then \cos \left ( \alpha +\beta \right )-\cos \left ( \alpha -\beta \right ) is equal to : 

  • Option 1)

     

    -1

     

     

     

  • Option 2)

     

    \sqrt{2}

  • Option 3)

     

    -\sqrt{2}

  • Option 4)

     

    0

Answers (1)

best_answer

 

Addition Formulae -

\cos \left ( A+B \right )= \cos A\cos B-\sin A\sin B

- wherein

A and B are two angles.

 

 

Subtraction Formulae -

\cos \left ( A-B \right )= \cos A\cos B+\sin A\sin B

- wherein

A and B are two angles.

AM\geq G.M.\\\\\frac{sin^{4}\alpha +4cos^{4}\beta +1+1}{4}\geq (4sin^{4}\alpha \times cos^{4}\beta )^{1/4}\\\\AM=GM\\\\\Rightarrow sin^{4}\alpha =4cos^{4}\beta =2\\\\sin^{4}\alpha =1\Rightarrow \alpha =\pi /2\\\\cos\beta =\frac{1}{\sqrt{2}}\Rightarrow \beta =\pi /4\\\\hence \: -2\: sin \: \alpha\: sin\: \beta =-1\times 1\times \frac{1}{\sqrt{2}} \\\\=-\sqrt{2}

 


Option 1)

 

-1

 

 

 

Option 2)

 

\sqrt{2}

Option 3)

 

-\sqrt{2}

Option 4)

 

0

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