Given:
The trigonometric expression is given as,
[tex]\sin 105\degree\cos 45\degree-\cos 105\degree\sin 45\degree[/tex]
The objective is to find the exact value of the expression.
Explanation:
Consider the general formula,
[tex]\sin a\cos b-\cos a\sin b=\sin (a+b)\ldots\text{.}(1)[/tex]
By comparing the given equation with the RHS of equation (1),
[tex]\begin{gathered} a=105\degree \\ b=45\degree \end{gathered}[/tex]
To find the value:
On plugging the obtained values of a and b in equation (1),
[tex]\sin 105\degree\cos 45\degree-\cos 105\degree\sin 45\degree=\sin (105-45)[/tex]
On further solving the above equation,
[tex]\sin 105\degree\cos 45\degree-\cos 105\degree\sin 45\degree=\sin (60)\degree[/tex]
From the trigonometric table,
[tex]\sin 60\degree=\frac{\sqrt[]{3}}{2}[/tex]
Hence, the exact value of the expression is √3/2.
