Evaluate Trigonometric Expression

To find the value of the given trigonometric expression, we substitute the known values of trigonometric functions for standard angles and simplify.

Identify Standard Trigonometric Values

The standard values required for the calculation are:

• $\sin 45^{\circ} = \frac{1}{\sqrt{2}}$
• $\sec 45^{\circ} = \frac{1}{\cos 45^{\circ}} = \sqrt{2}$
• $\tan 45^{\circ} = 1$
• $\cos 60^{\circ} = \frac{1}{2}$
• $\sin 30^{\circ} = \frac{1}{2}$
• $\sin 90^{\circ} = 1$

Calculate Expression Value

The expression is: $\sin 45^{\circ} \sec 45^{\circ} \tan 45^{\circ} + \cos 60^{\circ} \sin 30^{\circ} \sin 90^{\circ} - 4\cos 60^{\circ}$

Substitute the values into the expression:

$ \left( \frac{1}{\sqrt{2}} \times \sqrt{2} \times 1 \right) + \left( \frac{1}{2} \times \frac{1}{2} \times 1 \right) - \left( 4 \times \frac{1}{2} \right) $

Simplify each part of the expression:

• Part 1: $\frac{1}{\sqrt{2}} \times \sqrt{2} \times 1 = 1$
• Part 2: $\frac{1}{2} \times \frac{1}{2} \times 1 = \frac{1}{4}$
• Part 3: $4 \times \frac{1}{2} = 2$

Combine the simplified parts according to the original expression:

$ 1 + \frac{1}{4} - 2 $

Perform the final arithmetic operations:

$ \frac{4}{4} + \frac{1}{4} - \frac{8}{4} = \frac{4 + 1 - 8}{4} = \frac{5 - 8}{4} = \frac{-3}{4} $

The value of the expression is $\frac{-3}{4}$.