\begin{tabular}{l|l}2 & $$ \begin{array}{l} ext { Sonder om 'n sakrekenaar te gebruik , bepaal die waarde van: } \ \sin 45^{\circ} \cdot \cos ^{2} 60^{\circ}- an ^{2} 45^{\circ}\end{array} $$ \ \hline\end{tabular}

Question

\begin{tabular}{l|l}2 & $$ \begin{array}{l}	ext { Sonder om 'n sakrekenaar te gebruik , bepaal die waarde van: } \ \sin 45^{\circ} \cdot \cos ^{2} 60^{\circ}-	an ^{2} 45^{\circ}\end{array} $$ \ \hline\end{tabular}

UpStudy AI · Accepted Answer

To solve the expression $$ \sin 45^{\circ} \cdot \cos^{2} 60^{\circ} - \tan^{2} 45^{\circ} $$, we will evaluate each trigonometric function step by step.

1. **Calculate $$ \sin 45^{\circ} $$**:
   $$
   \sin 45^{\circ} = \frac{\sqrt{2}}{2}
   $$

2. **Calculate $$ \cos 60^{\circ} $$**:
   $$
   \cos 60^{\circ} = \frac{1}{2}
   $$

3. **Calculate $$ \cos^{2} 60^{\circ} $$**:
   $$
   \cos^{2} 60^{\circ} = \left(\frac{1}{2}\right)^{2} = \frac{1}{4}
   $$

4. **Calculate $$ \tan 45^{\circ} $$**:
   $$
   \tan 45^{\circ} = 1
   $$

5. **Calculate $$ \tan^{2} 45^{\circ} $$**:
   $$
   \tan^{2} 45^{\circ} = 1^{2} = 1
   $$

Now we can substitute these values back into the original expression:

$$
\sin 45^{\circ} \cdot \cos^{2} 60^{\circ} - \tan^{2} 45^{\circ} = \left(\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{1}{4}\right) - 1
$$

Calculating the first part:

$$
\frac{\sqrt{2}}{2} \cdot \frac{1}{4} = \frac{\sqrt{2}}{8}
$$

Now substituting this back into the expression:

$$
\frac{\sqrt{2}}{8} - 1
$$

To express this with a common denominator:

$$
\frac{\sqrt{2}}{8} - \frac{8}{8} = \frac{\sqrt{2} - 8}{8}
$$

Thus, the final value of the expression is:

$$
\frac{\sqrt{2} - 8}{8}
$$
The value of the expression $$ \sin 45^{\circ} \cdot \cos^{2} 60^{\circ} - \tan^{2} 45^{\circ} $$ is $$ \frac{\sqrt{2} - 8}{8} $$.

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