If `tantheta=4/3` then find the value of `(sintheta+costheta)/(sintheta-costheta)`

Answer

To solve the problem, we need to find the value of \((\sin \theta + \cos \theta) / (\sin \theta - \cos \theta)\) given that \(\tan \theta = \frac{4}{3}\).

**Step 1: Understand the relationship between sine, cosine, and tangent.**
Given that \(\tan \theta = \frac{4}{3}\), we can interpret this as the ratio of the opposite side (perpendicular) to the adjacent side (base) in a right-angled triangle.

**Hint:** Recall that \(\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}\).

**Step 2: Assign values to the sides of the triangle.**
Let the opposite side (perpendicular) be 4 and the adjacent side (base) be 3.

**Hint:** Use the values given in the tangent ratio to define the sides of the triangle.

**Step 3: Calculate the hypotenuse using the Pythagorean theorem.**
Using the Pythagorean theorem:
\[ \text{Hypotenuse} = \sqrt{(\text{Opposite})^2 + (\text{Adjacent})^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5. \]

**Hint:** Remember the Pythagorean theorem: \(c^2 = a^2 + b^2\).

**Step 4: Find the values of sine and cosine.**
Now we can find \(\sin \theta\) and \(\cos \theta\):
\[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{4}{5}, \]
\[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{3}{5}. \]

**Hint:** Use the definitions of sine and cosine based on the triangle sides.

**Step 5: Substitute the values into the expression.**
Now we substitute \(\sin \theta\) and \(\cos \theta\) into the expression \((\sin \theta + \cos \theta) / (\sin \theta - \cos \theta)\):
\[ \frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta} = \frac{\frac{4}{5} + \frac{3}{5}}{\frac{4}{5} - \frac{3}{5}}. \]

**Hint:** Combine the fractions carefully.

**Step 6: Simplify the expression.**
The numerator becomes:
\[ \frac{4}{5} + \frac{3}{5} = \frac{4 + 3}{5} = \frac{7}{5}. \]
The denominator becomes:
\[ \frac{4}{5} - \frac{3}{5} = \frac{4 - 3}{5} = \frac{1}{5}. \]
Thus, we have:
\[ \frac{\frac{7}{5}}{\frac{1}{5}}. \]

**Hint:** When dividing fractions, multiply by the reciprocal.

**Step 7: Final calculation.**
This simplifies to:
\[ \frac{7}{5} \times \frac{5}{1} = 7. \]

**Final Answer:** The value of \(\frac{\sin \theta + \cos \theta}{\sin \theta - \cos \theta}\) is \(7\).