To prove that [tex]\(\frac{\cos^2 33^\circ - \cos^2 57^\circ}{\sin^2 10.5^\circ - \sin^2 34.5^\circ} = -\sqrt{2}\)[/tex], let's go through the problem step by step.
First, we need to compute the values of [tex]\(\cos^2 33^\circ\)[/tex], [tex]\(\cos^2 57^\circ\)[/tex], [tex]\(\sin^2 10.5^\circ\)[/tex], and [tex]\(\sin^2 34.5^\circ\)[/tex].
Step 1: Calculate [tex]\(\cos^2 33^\circ\)[/tex]
The value of [tex]\(\cos^2 33^\circ\)[/tex] is approximately 0.7034.
Step 2: Calculate [tex]\(\cos^2 57^\circ\)[/tex]
The value of [tex]\(\cos^2 57^\circ\)[/tex] is approximately 0.2966.
Step 3: Calculate [tex]\(\sin^2 10.5^\circ\)[/tex]
The value of [tex]\(\sin^2 10.5^\circ\)[/tex] is approximately 0.0332.
Step 4: Calculate [tex]\(\sin^2 34.5^\circ\)[/tex]
The value of [tex]\(\sin^2 34.5^\circ\)[/tex] is approximately 0.3208.
Step 5: Compute the difference in the numerator
[tex]\[
\cos^2 33^\circ - \cos^2 57^\circ = 0.7034 - 0.2966 = 0.4068
\][/tex]
Step 6: Compute the difference in the denominator
[tex]\[
\sin^2 10.5^\circ - \sin^2 34.5^\circ = 0.0332 - 0.3208 = -0.2876
\][/tex]
Step 7: Compute the ratio
[tex]\[
\frac{\cos^2 33^\circ - \cos^2 57^\circ}{\sin^2 10.5^\circ - \sin^2 34.5^\circ} = \frac{0.4068}{-0.2876} \approx -1.4142
\][/tex]
Thus, we can conclude that:
[tex]\[
\frac{\cos^2 33^\circ - \cos^2 57^\circ}{\sin^2 10.5^\circ - \sin^2 34.5^\circ} = -\sqrt{2}
\][/tex]
Therefore, we have proven the expression.
