# Question 06769

Oct 15, 2017

$L H S = \sin \left(420\right) \cos \left(330\right) + \cos \left(300\right) \sin \left(- 330\right)$

$= \sin \left(4 \cdot 90 + 60\right) \cos \left(4 \cdot 90 - 30\right) + \cos \left(- 4 \cdot 90 + 60\right) \sin \left(- 4 \cdot 90 + 30\right)$

$= \sin 60 \cos 30 + \cos 60 \sin 30$

$= \sin \left(60 + 30\right) = \sin 90 = 1 = R H S$

Oct 15, 2017

Proved sin(420) cos(390) + cos(-300) sin(-330) = 1

#### Explanation:

We have to prove sin(420) cos(390) + cos(-300) sin(-330) = 1

Left Hand Side (L.H.S.)
$S \in \left(4. 90 + 60\right) . \cos \left(4. 90 + 30\right) + \cos \left(300\right) \left[- \sin \left(330\right)\right]$
[ As cos (-theta) = cos theta and sin(-theta)= - sin theta]#

$\Rightarrow \sin 60 \cos 30 - \cos \left(4. 90 - 60\right) \sin \left(4. 90 - 30\right)$

$\Rightarrow \sin 60 \cos 30 - \cos 60 \left(- \sin 30\right)$

$\Rightarrow \sin 60 \cos 30 + \cos 60 \sin 30$

$\Rightarrow \sin \left(60 + 30\right)$ [ As sin(a+b) = sin a cos b + cos a sin b]

$\Rightarrow \sin 90$

$\Rightarrow 1$ = R.H.S.