To solve the equation \( \cos 35^{\circ} \cos 25^{\circ} - \sin 20^{\circ} \sin 10^{\circ} = \frac{1}{4} (\sqrt{3} + 1) \), we'll break it down step-by-step.
### Step 1: Understanding Trigonometric Identities
We know from trigonometric identities that:
[tex]\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \][/tex]
Let's apply this identity to the equation provided.
### Step 2: Applying the Trigonometric Identity
Rewrite the left-hand side of the equation in terms of trigonometric identity:
[tex]\[ \cos 35^{\circ} \cos 25^{\circ} - \sin 20^{\circ} \sin 10^{\circ} \][/tex]
We notice that \( \cos 35^{\circ} \cos 25^{\circ} - \sin 35^{\circ} \sin 25^{\circ} \) is equivalent to \( \cos (35^{\circ} + 25^{\circ}) \). But in our case, it's important to recognize that \( 35^\circ + 25^\circ=60^\circ \), so:
[tex]\[ \cos (35^\circ + 25^\circ) = \cos 60^\circ \][/tex]
Similarly,
[tex]\[ \sin 20^\circ \sin 10^\circ \][/tex]
does not directly simplify using standard trigonometric sums, so we must retain the expression \( \sin 20^\circ \sin 10^\circ \).
But in our scenario, we look at the final result through numerical comparison.
### Step 3: Calculation and Comparison
Compute the values of:
[tex]\[ \cos 35^\circ \cos 25^\circ - \sin 20^\circ \sin 10^\circ \][/tex]
Numerically evaluated, it results in:
[tex]\[ 0.6830127018922194 \][/tex]
### Step 4: Right-Hand Side Evaluation
Now, evaluate the right-hand side of the given equation:
[tex]\[ \frac{1}{4} (\sqrt{3} + 1) \][/tex]
Numerically, this results in:
[tex]\[ 0.6830127018922193 \][/tex]
### Step 5: Comparison
We observe that both sides of the equation are very close numerically:
[tex]\[ \cos 35^\circ \cos 25^\circ - \sin 20^\circ \sin 10^\circ \approx \frac{1}{4} (\sqrt{3} + 1) \][/tex]
Both sides result in:
[tex]\[ 0.683 \][/tex]
Thus, the equation [tex]\( \cos 35^{\circ} \cos 25^{\circ} - \sin 20^{\circ} \sin 10^{\circ} = \frac{1}{4} (\sqrt{3} + 1) \)[/tex] is verified to be true based on our numerical calculations.
