If A = $45^{\circ},$ verify that:

(i) $s i n 2 A = 2 s i n A c o s A$ (ii) $c o s 2 A = 2 c o s^{2} A - 1 = 1 - 2 s i n^{2} A$

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Solution

(i) $A = 45^{o}$

$S i n 2 A = 2 s i n A c o s A$

$S i n 2 (45^{o}) = 2 s i n 45^{o} C o s 45^{o}$

$S i n 90^{o} = 2 S i n 45^{o} C o s 45^{o}$

$1 = 2 \times \frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}}$

$1 = 1$

Hence proved.

(ii) $A = 45^{o}$

to prove, $c o s 2 A = 2 c o s^{2} A - 1 = 1 - s i n^{2} A$

$c o s 2 A = c o s 2 \times 45^{o} = c o s 90^{o} = 0$

$2 c o s^{2} A - 1 = 2 \times (\frac{1}{\sqrt{2}})^{2} - 1 = 2 \times \frac{1}{2} - 1 = 1 - 1 = 0$

$1 - s i n^{2} A = 1 - (s i n 90^{o}) = 1 - 1 = 0$

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