The value of cos 105° + sin 105° is

Using, cos (x + y) = cos x cos y - sin x sin y

and sin (x + y) = sin x cos y + cos x sin y

Expression : cos 105° + sin 105°

= cos (60 + 45) + sin (60 + 45)

= (cos 60.cos 45 - sin 60.sin 45) + (sin 60.cos 45 + cos 60.sin 45)

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

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

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

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