Respuesta :

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.

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