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{\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.