If         is the arithmetic mean of  and , then   is  equal to : Option 1) Option 2) Option 3) Option 4)

D Divya Saini

As we learnt in

Trigonometric Ratios of Submultiples of an Angle -

- wherein

This shows the formulae for half angles and their doubles.

$cos\alpha +cos\beta = \frac{3}{2}$    $\Rightarrow 2 cos\frac{\alpha +\beta }{2} cos \frac{\alpha -\beta }{2 } = \frac{3}{2}\ \: \: \: \: \: \: \: \: \: ................(1)$

$sin\alpha +sin\beta = \frac{1}{2}$   $\Rightarrow 2 sin\frac{\alpha +\beta }{2}\:\:cos\frac{\alpha -\beta }{2}=\frac{1}{2}\ \, \, \, \, \, \, \, \, .........(2)$

dividing  (2) by (1), we get

$tan\frac{\left ( \alpha +\beta \right )}{2}=\frac{1}{3}$

$\Rightarrow tan\left ( \alpha +\beta \right )=\frac{2\times \frac{1}{3}}{1-\frac{1}{9}}=\frac{3}{4}$

$\Rightarrow sin \left ( \alpha +\beta \right )=\frac{3}{5},\:\:\:cos\left ( \alpha +\beta \right ) = \frac{4}{5}$

Thus, $sin \left ( \alpha +\beta \right )+cos\left ( \alpha +\beta \right )=\frac{7}{5}$

$\Rightarrow sin2\theta +cos2\theta =\frac{7}{5}$

Option 1)

This option is incorrect.

Option 2)

This option is incorrect.

Option 3)

This option is correct.

Option 4)

This option is incorrect.

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