Math Problem Statement

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

Solution

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

  • ∠B = 90° (so, triangle ABC is a right-angled triangle).
  • tanA=13\tan A = \frac{1}{\sqrt{3}}.

We need to show that:

cosA1tanA+sinA1cotA=sinA+cosA\frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A} = \sin A + \cos A

Step 1: Use the trigonometric identity for tanA\tan A

We know that:

tanA=sinAcosA\tan A = \frac{\sin A}{\cos A}

It is given that:

tanA=13\tan A = \frac{1}{\sqrt{3}}

So, we can find sinA\sin A and cosA\cos A. Since tanA=sinAcosA=13\tan A = \frac{\sin A}{\cos A} = \frac{1}{\sqrt{3}}, we know that:

sinA=12,cosA=32\sin A = \frac{1}{2}, \quad \cos A = \frac{\sqrt{3}}{2}

Step 2: Find cotA\cot A

cotA=1tanA=3\cot A = \frac{1}{\tan A} = \sqrt{3}

Step 3: Substitute sinA\sin A, cosA\cos A, tanA\tan A, and cotA\cot A into the equation

We need to evaluate:

cosA1tanA+sinA1cotA\frac{\cos A}{1 - \tan A} + \frac{\sin A}{1 - \cot A}

Substituting the known values of sinA\sin A, cosA\cos A, tanA\tan A, and cotA\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.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Right-angled triangles
Trigonometric identities

Formulas

tan A = sin A / cos A
cot A = 1 / tan A
Basic trigonometric identities

Theorems

Relationship between sine, cosine, and tangent in right-angled triangles
Trigonometric simplification

Suitable Grade Level

Grades 9-12