JEE MAIN - Mathematics (2020 - 5th September Evening Slot - No. 11)
If L = sin2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$ and
M = cos2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$, then :
M = cos2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$, then :
L = $$ - {1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}$$
M = $${1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}$$
M = $${1 \over {4\sqrt 2 }} + {1 \over 4}\cos {\pi \over 8}$$
L = $${1 \over {4\sqrt 2 }} - {1 \over 4}\cos {\pi \over 8}$$
Explanation
We will use here those two formulas,
sin2 $$\theta $$ = $${{1 - \cos 2\theta } \over 2}$$ and cos2 $$\theta $$ = $${{1 + \cos 2\theta } \over 2}$$
L = sin2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$
$$ \Rightarrow $$ L = $$\left( {{{1 - \cos \left( {{\pi \over 8}} \right)} \over 2}} \right)$$ - $$\left( {{{1 - \cos \left( {{\pi \over 4}} \right)} \over 2}} \right)$$
$$ \Rightarrow $$ L = $${1 \over 2}\left( {\cos \left( {{\pi \over 4}} \right) - \cos \left( {{\pi \over 8}} \right)} \right)$$
$$ \Rightarrow $$ L = $${1 \over {2\sqrt 2 }} - {1 \over 2}\cos \left( {{\pi \over 8}} \right)$$
M = cos2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$
$$ \Rightarrow $$ M = $$\left( {{{1 + \cos \left( {{\pi \over 8}} \right)} \over 2}} \right)$$ - $$\left( {{{1 - \cos \left( {{\pi \over 4}} \right)} \over 2}} \right)$$
$$ \Rightarrow $$ M = $${1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}$$
sin2 $$\theta $$ = $${{1 - \cos 2\theta } \over 2}$$ and cos2 $$\theta $$ = $${{1 + \cos 2\theta } \over 2}$$
L = sin2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$
$$ \Rightarrow $$ L = $$\left( {{{1 - \cos \left( {{\pi \over 8}} \right)} \over 2}} \right)$$ - $$\left( {{{1 - \cos \left( {{\pi \over 4}} \right)} \over 2}} \right)$$
$$ \Rightarrow $$ L = $${1 \over 2}\left( {\cos \left( {{\pi \over 4}} \right) - \cos \left( {{\pi \over 8}} \right)} \right)$$
$$ \Rightarrow $$ L = $${1 \over {2\sqrt 2 }} - {1 \over 2}\cos \left( {{\pi \over 8}} \right)$$
M = cos2$$\left( {{\pi \over {16}}} \right)$$ - sin2$$\left( {{\pi \over {8}}} \right)$$
$$ \Rightarrow $$ M = $$\left( {{{1 + \cos \left( {{\pi \over 8}} \right)} \over 2}} \right)$$ - $$\left( {{{1 - \cos \left( {{\pi \over 4}} \right)} \over 2}} \right)$$
$$ \Rightarrow $$ M = $${1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}$$
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