The expression provided in the image is:

$\frac{\cos 77^\circ}{\sin 13^\circ} + \cos^2 52^\circ + \cos^2 38^\circ - 3 \tan^2 45^\circ$

Step 1: Trigonometric identities and simplifications

1. Notice that:
   • $\cos 77^\circ = \sin 13^\circ$ (since $\cos \theta = \sin (90^\circ - \theta)$).
   • $\tan 45^\circ = 1$, so $\tan^2 45^\circ = 1$.

Now let's simplify each part of the expression:

$\frac{\cos 77^\circ}{\sin 13^\circ} = \frac{\sin 13^\circ}{\sin 13^\circ} = 1$

Next, we can evaluate:

$\cos^2 52^\circ + \cos^2 38^\circ$

Since $\cos^2 \theta + \cos^2 (90^\circ - \theta) = 1$, we have:

$\cos^2 52^\circ + \cos^2 38^\circ = 1$

Finally, the term involving $\tan^2 45^\circ$ is:

$3 \cdot \tan^2 45^\circ = 3 \cdot 1 = 3$

Step 2: Putting it all together

Now sum up all the terms:

$1 + 1 - 3 = -1$

Final Answer:

The value of the given expression is $-1$.

Would you like more details or have further questions?

Here are some related questions for practice:

1. What is the value of $\tan(45^\circ)$ and why is it significant?
2. How does the identity $\cos \theta = \sin (90^\circ - \theta)$ simplify trigonometric expressions?
3. Evaluate $\cos^2(30^\circ) + \sin^2(30^\circ)$ and explain the result.
4. Can you apply the identity $\cos^2 \theta + \sin^2 \theta = 1$ to different angles?
5. What is the significance of the angle $45^\circ$ in trigonometry?

Tip: Always check for identities like $\cos \theta = \sin(90^\circ - \theta)$ in simplifying trigonometric expressions!