Tính giá trị biểu thức

To calculate the value of the expression:

\[
\cos^2 26^\circ - \frac{10 \sin 55^\circ}{\cos 35^\circ} + \tan 39^\circ + \sin^2 26^\circ - \cot 51^\circ
\]

we can simplify it step by step.

1. **Using Trigonometric Identities**:
- \(\sin 55^\circ = \cos 35^\circ\) (since \(55^\circ + 35^\circ = 90^\circ\))
- \(\cot 51^\circ = \frac{1}{\tan 51^\circ}\) and \(\tan 51^\circ = \cot 39^\circ\) (since \(51^\circ + 39^\circ = 90^\circ\))

2. **Substituting Values**:
- \(\cos^2 26^\circ + \sin^2 26^\circ = 1\)
- Therefore, \(\cos^2 26^\circ + \sin^2 26^\circ = 1\)

3. **Simplifying the Expression**:
- The expression becomes:
\[
1 - \frac{10 \cos 35^\circ}{\cos 35^\circ} + \tan 39^\circ - \cot 39^\circ
\]
- This simplifies to:
\[
1 - 10 + \tan 39^\circ - \cot 39^\circ
\]
- Which further simplifies to:
\[
-9 + \tan 39^\circ - \cot 39^\circ
\]

4. **Calculating \(\tan 39^\circ - \cot 39^\circ\)**:
- \(\tan 39^\circ - \cot 39^\circ = \tan 39^\circ - \frac{1}{\tan 39^\circ}\)
- Let \(x = \tan 39^\circ\), then:
\[
x - \frac{1}{x} = \frac{x^2 - 1}{x}
\]

5. **Final Calculation**:
- The final expression becomes:
\[
-9 + \frac{x^2 - 1}{x}
\]
- To find the numerical value, you can calculate \(\tan 39^\circ\) using a calculator and substitute it back into the expression.

This will give you the final numerical value of the expression.