The mathematical expression in the image is:

$\frac{1 - 4\sin(10^\circ) \cdot \sin(70^\circ)}{2\cos(10^\circ) + 2}.$

Solution:

Step 1: Simplify $\sin(10^\circ) \cdot \sin(70^\circ)$

Using the trigonometric product-to-sum formula: $\sin(A) \sin(B) = \frac{1}{2} \left[\cos(A-B) - \cos(A+B)\right],$ we substitute $A = 10^\circ$ and $B = 70^\circ$: $\sin(10^\circ) \cdot \sin(70^\circ) = \frac{1}{2} \left[\cos(10^\circ - 70^\circ) - \cos(10^\circ + 70^\circ)\right].$

Simplify the angles: $\sin(10^\circ) \cdot \sin(70^\circ) = \frac{1}{2} \left[\cos(-60^\circ) - \cos(80^\circ)\right].$

Using $\cos(-\theta) = \cos(\theta)$, we get: $\sin(10^\circ) \cdot \sin(70^\circ) = \frac{1}{2} \left[\cos(60^\circ) - \cos(80^\circ)\right].$

Substitute $\cos(60^\circ) = \frac{1}{2}$: $\sin(10^\circ) \cdot \sin(70^\circ) = \frac{1}{2} \left[\frac{1}{2} - \cos(80^\circ)\right].$

Simplify: $\sin(10^\circ) \cdot \sin(70^\circ) = \frac{1}{4} - \frac{1}{2}\cos(80^\circ).$

Step 2: Substitute back into the numerator

The numerator becomes: $1 - 4\left(\frac{1}{4} - \frac{1}{2}\cos(80^\circ)\right).$

Simplify: $1 - 4\left(\frac{1}{4}\right) + 4\left(\frac{1}{2}\cos(80^\circ)\right) = 1 - 1 + 2\cos(80^\circ).$

So the numerator is: $2\cos(80^\circ).$

Step 3: Simplify the denominator

The denominator is: $2\cos(10^\circ) + 2 = 2\left(\cos(10^\circ) + 1\right).$

Step 4: Write the simplified expression

The simplified expression becomes: $\frac{2\cos(80^\circ)}{2\left(\cos(10^\circ) + 1\right)}.$

Cancel the factor of 2: $\frac{\cos(80^\circ)}{\cos(10^\circ) + 1}.$

Final Answer:

$\boxed{\frac{\cos(80^\circ)}{\cos(10^\circ) + 1}}$

Would you like further details or have additional questions?

Related Questions:

1. How do product-to-sum trigonometric identities work?
2. Why is $\cos(-\theta) = \cos(\theta)$?
3. How can $\cos(80^\circ)$ be expressed in terms of $\sin(10^\circ)$?
4. What are the common trigonometric simplifications for small angles like $10^\circ$ or $80^\circ$?
5. How does the numerator simplify step-by-step in this case?

Tip:

Always check if trigonometric identities (like product-to-sum or sum-to-product) can simplify complex products or sums of sine and cosine!