https://r2.erweima.ai/mathbot/image/1724850143444ofki8ksq.jpg

Let's go through each of the trigonometric identities and equations in the image to solve them.

### Problem (1):
\[
1 + \tan^2 x = \sec^2 x
\]
This is a well-known trigonometric identity. It is one of the Pythagorean identities:
\[
1 + \tan^2 x = \sec^2 x
\]
So, this equation is **correct**.

### Problem (2):
\[
\cot^2 x + 1 = \csc^2 x
\]
This is another Pythagorean identity for cotangent and cosecant. The identity is:
\[
\cot^2 x + 1 = \csc^2 x
\]
So, this equation is also **correct**.

### Problem (3):
\[
\frac{1 + \tan^2 x}{\sec x} \cdot \cos x = \sec x
\]
Let's simplify the left-hand side:

1. Use the identity \( 1 + \tan^2 x = \sec^2 x \):
\[
\frac{\sec^2 x}{\sec x} \cdot \cos x
\]
2. Simplify \( \frac{\sec^2 x}{\sec x} = \sec x \):
\[
\sec x \cdot \cos x
\]
3. Since \( \sec x = \frac{1}{\cos x} \), we get:
\[
\frac{1}{\cos x} \cdot \cos x = 1
\]
This simplifies to 1, so the equation becomes:
\[
1 = \sec x
\]
Therefore, this is **incorrect** unless \( x = 0 \), where \( \sec(0) = 1 \).

### Problem (4):
\[
\sin x + \cos x = \frac{1}{\tan x}
\]
To check this, let's rewrite the right-hand side in terms of sine and cosine:
\[
\frac{1}{\tan x} = \frac{1}{\frac{\sin x}{\cos x}} = \frac{\cos x}{\sin x}
\]
So the equation becomes:
\[
\sin x + \cos x = \frac{\cos x}{\sin x}
\]
This equation does not hold in general, so this is **incorrect**.

### Problem (5):
\[
(1 + \sin x)(1 - \sin x) = \cos^2 x
\]
This is a factored form of a difference of squares:
\[
(1 + \sin x)(1 - \sin x) = 1 - \sin^2 x
\]
Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \), we get:
\[
1 - \sin^2 x = \cos^2 x
\]
So, this equation is **correct**.

### Problem (6):
\[
\cot x \cdot \sin x = 1
\]
Rewrite cotangent in terms of sine and cosine:
\[
\cot x = \frac{\cos x}{\sin x}
\]
So the equation becomes:
\[
\frac{\cos x}{\sin x} \cdot \sin x = \cos x
\]
This simplifies to:
\[
\cos x = 1
\]
This is true when \( x = 0 \), but not generally true for all values of \( x \), so this equation is **incorrect**.

---

Would you like more details on any of these problems or their solutions? Here are some follow-up questions you might find helpful:

1. What are the Pythagorean identities in trigonometry, and how are they derived?
2. Can you prove that \( \frac{1 + \tan^2 x}{\sec x} \cdot \cos x = \sec x \) only holds for certain angles?
3. How can the identity \( (1 + \sin x)(1 - \sin x) = \cos^2 x \) be applied in integration problems?
4. Why does the equation \( \sin x + \cos x = \frac{1}{\tan x} \) fail for general values of \( x \)?
5. In what situations is the equation \( \cot x \cdot \sin x = 1 \) valid?

**Tip**: Always check the domain and range of trigonometric functions to see if an equation holds for all values of the variable.

Explore how to solve various trigonometric identity problems step-by-step. Includes detailed explanations and solutions for equations involving tan, sec, cot, csc, sin, and cos. Suitable for high school students and anyone preparing for trigonometry exams.

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