Question

cos 225+sin 165= pls give me solution
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Answer 1

Answer:

$\frac{\sqrt{2}}{4}(\sqrt{3}-1)$

Step-by-step explanation:

Here, the given expression is,

$cos 225^{\circ} + sin 165^{\circ}$

$=cos ( 270 - 45)^{\circ} + sin (90 + 75)$

$= -sin 45^{\circ} + sin 75^{\circ}$

( Since, cos (270° - x) = - sin x and sin (90 + x) = sin x )

$= -\frac{1}{\sqrt{2}}+ sin (45 + 30)^{\circ}$

$= -\frac{1}{\sqrt{2}} + sin 45^{\circ}\times cos 30^{\circ }+ cos 45^{\circ}\times sin 30^{\circ}$

( Since, sin(a+b) = sin a × cos b + cos a × sin b )

$= -\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}+ \frac{1}{\sqrt{2}}\times \frac{1}{2}$

$=-\frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}}$

$=\frac{-2+\sqrt{3}+1}{2\sqrt{2}}$

$=\frac{\sqrt{3}-1}{2\sqrt{2}}$

$=\frac{\sqrt{2}}{4}(\sqrt{3}-1)$