Question

Prove :


$1) sin \: 50 \degree + sin \: 70 \degree \: = \sqrt{3} cos \: 10 \degree$
$2)1 - sin2a = 2sin^{2} (45 \degree - a)$

Answer 1

sin^2 5 + sin^2 10 + sin^2 15 + …… + sin^2 75 + sin ^2 80 +sin^2 85 + sin^2 90

(sin^2 5 + sin^2 85 )+( sin^2 10 + sin^2 80) +( sin^2 15 + sin^2 75) + ……..+ sin ^2 45 + sin^2 90

(sin^2 5 + cos^2 5 )+( sin^2 10 + cos^2 10) +( sin^2 15 + cos^2 15) + ……..+ sin ^2 45 + sin^2 90

(1 + 1 + 1 +1 + 1 + 1 +1 + 1 ) + 1/2 +1

19/2

Answer 2

$\sf{1) \: Sin \: 50 \degree + Sin \: 70 \degree=\sqrt{3} \: Cos \: 10 \degree}$

$\texttt{\underline{Solution}:}$

$\sf{ L.H.S = Sin \: 50 \degree + Sin \: 70 \degree}$

$\sf{ \implies 2 \: Sin ( \frac{70 \degree + \: 50 \degree}{2} }) \: . \: Cos( \frac{70 \degree - \: 50 \degree}{2} )$

$\sf{ \implies 2 \: Sin( \frac{ 120 \degree}{2}) \: . \: Cos( \frac{20 \degree}{2} )}$

$\sf{ \implies 2 \: Sin \: 60 \degree \: . \: Cos \: 10 \degree}$

$\sf{ \implies 2 \times \frac{ \sqrt{3} }{2} \: . \: Cos \: 10 \degree}$

$\sf{ \implies \sqrt{3} \: Cos \:10 \degree} \: R.H.S \: Proved.$

$\pink{\boxed{ \mathcal \green{ \therefore \: L.H.S = R.H.S \: Proved.}}}$

$\sf{2) \: 1 -Sin \: 2A = {2 \: Sin}^{2} (45 \degree -A)}$

$\texttt{\underline{Solution}:}$

$\sf{R.H.S = {2 \: Sin}^{2} (45 \degree -A)}$

$\sf{ \implies 1 - Cos \: 2 \: (45 \degree - A})$

$\sf{ \implies 1 - Cos \:(90 \degree - 2A})$

$\sf{ \implies 1 - Sin \: 2A} \: \: L.H.S \: Proved.$

$\pink{\boxed{ \mathcal \green{ \therefore \: L.H.S = R.H.S \: Proved.}}}$