Evaluate cos105degree+sin105degree

The value of cos 105^o + sin 105^o is {tex}\frac{1}{\sqrt{2}}{/tex}
Cos trigonometric values are positive in “first and fourth” quadrants in coordinate axes.
Sin trigonometric values are positive in “first and second” quadrants in coordinate axes.
cos 105^o + sin 105^o = cos (90^o + 15^o) + sin (90^o + 15^o)
= - sin (15^o) + cos (15^o) (ascos (90^o + {tex}\theta{/tex}) = - sin {tex}\theta{/tex} and sin (90^o + {tex}\theta{/tex}) = cos {tex}\theta{/tex})
= cos 15^o - sin 15^o
{tex}\left.=\sqrt{2}\left[\frac{1}{\sqrt{2}} \cos 15^{\circ}-\frac{1}{\sqrt{2}} \sin 15^{\circ}\right] \quad \text { (Multiply and divide by } \sqrt{2}\right){/tex}
{tex}=\sqrt{2}\left[\sin 45^{\circ} \cos 15^{\circ}-\cos 45^{\circ} \sin 15^{\circ}\right] \quad\left(A s \sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right){/tex}
{tex}=\sqrt{2}\left[\sin \left(45^{\circ}-15^{\circ}\right)\right] \quad(A \sin (A-B)=\sin A \cos B-\cos A \sin B){/tex}
{tex}=\sqrt{2}\left[\sin 30^{\circ}\right]{/tex}
{tex}=\sqrt{2} \times \frac{1}{2}=\frac{1}{\sqrt{2}}{/tex}