Question: take your time and solve each questions while explaining every step so that i can understand as if I'm a baby and order them very well
1. **Problem:** Determine which of the following trigonometric identities is not true.
Given options:
A. $\sin^2 x + \cos^2 x = 1$
B. $1 + \tan^2 x = \sec^2 x$
C. $1 + \cot^2 x = -\csc^2 x$
D. $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$
2. **Formula and rules:**
- Pythagorean identity: $\sin^2 x + \cos^2 x = 1$ is always true.
- Another Pythagorean identity: $1 + \tan^2 x = \sec^2 x$ is true.
- The identity involving cotangent and cosecant is $1 + \cot^2 x = \csc^2 x$, note the positive sign.
- The tangent subtraction formula is $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$.
3. **Step-by-step check:**
- Option A matches the Pythagorean identity and is true.
- Option B matches the second Pythagorean identity and is true.
- Option C has a negative sign before $\csc^2 x$, which is incorrect; it should be positive.
- Option D matches the tangent subtraction formula and is true.
4. **Conclusion:**
Option C is not true.
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**Final answer:** Option C is not true.
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