use one or more of the six sum and difference identities to find the exact value of the expression…

Explanation:

Step1: Decompose the angle

Decompose (105^{\circ}) into (60^{\circ}+45^{\circ}). So, (\sin(105^{\circ})=\sin(60^{\circ} + 45^{\circ})).

Step2: Apply the sine sum identity

The sine sum identity is (\sin(A + B)=\sin A\cos B+\cos A\sin B). Here (A = 60^{\circ}) and (B=45^{\circ}). [ \begin{align*} \sin(60^{\circ}+45^{\circ})&=\sin60^{\circ}\cos45^{\circ}+\cos60^{\circ}\sin45^{\circ}\ \end{align*} ]

Step3: Substitute the values of trigonometric functions

We know that (\sin60^{\circ}=\frac{\sqrt{3}}{2}), (\cos45^{\circ}=\frac{\sqrt{2}}{2}), (\cos60^{\circ}=\frac{1}{2}), and (\sin45^{\circ}=\frac{\sqrt{2}}{2}). [ \begin{align*} \sin(60^{\circ}+45^{\circ})&=\frac{\sqrt{3}}{2}\times\frac{\sqrt{2}}{2}+\frac{1}{2}\times\frac{\sqrt{2}}{2}\ &=\frac{\sqrt{6}}{4}+\frac{\sqrt{2}}{4}\ &=\frac{\sqrt{6}+\sqrt{2}}{4} \end{align*} ]

Answer:

(\frac{\sqrt{6}+\sqrt{2}}{4})