Triangle ABC <b= 90° , tan A= 1/√3 . Show that  Cos A/1-tan A + sin A/1-Cot A = Sin A+ Cos A

We are given the following information about a right-angled triangle ABC:

- ∠B = 90° (so, triangle ABC is a right-angled triangle).
- \( \tan A = \frac{1}{\sqrt{3}} \).

We need to show that:

\[
\frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A} = \sin A + \cos A
\]

### Step 1: Use the trigonometric identity for \(\tan A\)
We know that:

\[
\tan A = \frac{\sin A}{\cos A}
\]

It is given that:

\[
\tan A = \frac{1}{\sqrt{3}}
\]

So, we can find \(\sin A\) and \(\cos A\). Since \(\tan A = \frac{\sin A}{\cos A} = \frac{1}{\sqrt{3}}\), we know that:

\[
\sin A = \frac{1}{2}, \quad \cos A = \frac{\sqrt{3}}{2}
\]

### Step 2: Find \(\cot A\)
\[
\cot A = \frac{1}{\tan A} = \sqrt{3}
\]

### Step 3: Substitute \(\sin A\), \(\cos A\), \(\tan A\), and \(\cot A\) into the equation

We need to evaluate:

\[
\frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A}
\]

Substituting the known values of \(\sin A\), \(\cos A\), \(\tan A\), and \(\cot A\):

1. For the first term:
\[
\frac{\cos A}{1 - \tan A} = \frac{\frac{\sqrt{3}}{2}}{1 - \frac{1}{\sqrt{3}}}
= \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3} - 1}{\sqrt{3}}}
= \frac{\frac{\sqrt{3}}{2} \cdot \sqrt{3}}{\sqrt{3} - 1}
= \frac{3}{2(\sqrt{3} - 1)}
\]

2. For the second term:
\[
\frac{\sin A}{1 - \cot A} = \frac{\frac{1}{2}}{1 - \sqrt{3}}
= \frac{\frac{1}{2}}{\frac{1 - \sqrt{3}}{1}}
= \frac{1}{2(1 - \sqrt{3})}
\]

### Step 4: Combine the two terms

Now, combine the two terms:

\[
\frac{3}{2(\sqrt{3} - 1)} + \frac{1}{2(1 - \sqrt{3})}
\]

Since \(1 - \sqrt{3} = -( \sqrt{3} - 1)\), we can simplify this expression as follows:

\[
\frac{3}{2(\sqrt{3} - 1)} - \frac{1}{2(\sqrt{3} - 1)} = \frac{2}{2(\sqrt{3} - 1)} = \frac{1}{\sqrt{3} - 1}
\]

Thus, this simplifies further to \(\sin A + \cos A\), as required.

### Conclusion:
We have shown that:

\[
\frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A} = \sin A + \cos A
\]

Would you like further clarification or details on any step?

Here are some related questions you might find interesting:
1. What are the basic trigonometric identities?
2. How is the value of sine and cosine derived for specific angles?
3. How do we derive the values of \(\tan\) and \(\cot\) for standard angles?
4. What are the properties of right-angled triangles with respect to trigonometry?
5. How does the tangent function relate to other trigonometric functions?

**Tip:** When working with trigonometric identities, always check if simplifying one side can help directly match the other.

Triangle ABC, ∠B = 90°, tan A = 1/√3. Show that Cos A/(1 - tan A) + Sin A/(1 - Cot A) = Sin A + Cos A.

Explore how to prove the trigonometric identity Cos A/(1 - tan A) + Sin A/(1 - Cot A) = Sin A + Cos A for triangle ABC, where ∠B = 90° and tan A = 1/√3. This detailed solution involves fundamental trigonometric concepts and is suitable for students in Grades 9-12.

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