Prove that: `sin^2 42^0-cos^2 78^0=(sqrt(5)+1)/8`

Explanation

To prove that \( \sin^2 42^\circ - \cos^2 78^\circ = \frac{\sqrt{5} + 1}{8} \), we can follow these steps:

### Step 1: Rewrite \( \cos^2 78^\circ \)
We know that \( \cos(90^\circ - \theta) = \sin(\theta) \). Therefore, we can rewrite \( \cos^2 78^\circ \) as:
\[ \cos^2 78^\circ = \sin^2(90^\circ - 78^\circ) = \sin^2 12^\circ \]
Thus, the expression becomes:
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