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