Which of the following is not true ?

Options

A. `sin cos^(-1)tan cot^(-1)x=sqrt(1-(1)/(x^(2)))`
B. `cos tan^(-1)cot sin^(-1)x = x`
C. `tan cot^(-1)sin cos^(-1)x = (1)/(sqrt(1-x^(2)))`
D. `cot sin^(-1)cos tan^(-1)x=sqrt(1-x^(2))`

Explanation

To solve the problem of determining which of the given statements is not true, we will analyze each option step by step.

### Step-by-Step Solution:

**Option 1: \( \sin(\cos^{-1}(10 \cot^{-1}(x))) \)**

1. **Use the identity**: \( \cot^{-1}(x) = \tan^{-1}\left(\frac{1}{x}\right) \).
- This gives us:
\[ \sin(\cos^{-1}(10 \tan^{-1}(\frac{1}{x}))) \]

2. **Apply the identity**: \( \cos^{-1}(y) = \sin^{-1}(\sqrt{1 - y^2}) \).
- Hence, we have:
\[ \sin(\sin^{-1}(\sqrt{1 - (10 \tan^{-1}(\frac{1}{x}))^2})) = \sqrt{1 - (10 \tan^{-1}(\frac{1}{x}))^2} \]
- This expression simplifies correctly, confirming that this option is true.

**Option 2: \( \cos(10^{-1}(\cot(\sin^{-1}(x)))) = x \)**

1. **Use the identity**: \( \sin^{-1}(x) = \cot^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right) \).
- This gives us:
\[ \cos(10^{-1}\left(\frac{\sqrt{1 - x^2}}{x}\right)) \]

2. **Apply the identity**: \( \tan^{-1}(y) = \cos^{-1}\left(\frac{1}{\sqrt{1+y^2}}\right) \).
- Thus, we have:
\[ \cos\left(\cos^{-1}\left(\frac{x}{\sqrt{x^2 + (1 - x^2)}}\right)\right) = x \]
- This expression holds true, confirming that this option is also true.

**Option 3: \( \tan(\cot^{-1}(\sin(\cos^{-1}(x)))) \)**

1. **Use the identity**: \( \cos^{-1}(x) = \sin^{-1}(\sqrt{1 - x^2}) \).
- This gives us:
\[ \tan(\cot^{-1}(\sin(\sin^{-1}(\sqrt{1 - x^2})))) \]

2. **Apply the identity**: \( \cot^{-1}(y) = \tan^{-1}\left(\frac{1}{y}\right) \).
- Thus, we have:
\[ \tan\left(\tan^{-1}\left(\frac{1}{\sqrt{1 - x^2}}\right)\right) = \frac{1}{\sqrt{1 - x^2}} \]
- This expression is also true.

**Option 4: \( \cot(\sin^{-1}(\cos(\tan^{-1}(x)))) \)**

1. **Use the identity**: \( \tan^{-1}(x) = \cos^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right) \).
- This gives us:
\[ \cot(\sin^{-1}(\cos(\cos^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)))) \]

2. **Apply the identity**: \( \sin^{-1}(y) = \cot^{-1}\left(\frac{\sqrt{1-y^2}}{y}\right) \).
- This leads to:
\[ \cot(\cot^{-1}(x)) = x \]
- However, the expression in the option states it equals \( \sqrt{1 - x^2} \), which is not true.

### Conclusion:
The statement that is not true is **Option 4**.