From a point on the ground the angle of elevation of top of a house and top of a chimney surmounted on the house are respectively `x^(@) and 45^(@)`. If height of the house is h meter then height of the chimney is

Options

A. a)`hcotx+h`
B. b)`hcotx-h`
C. c)`htanx-h`
D. d)`htanx+h`

Answer

To solve the problem, we will use the concepts of trigonometry, specifically the tangent function, which relates the angle of elevation to the height and distance from the object.

### Step-by-Step Solution:

1. **Understanding the Problem**:
- We have a house of height \( h \) meters.
- There is a chimney on top of the house.
- The angle of elevation to the top of the house from a point on the ground is \( x^\circ \).
- The angle of elevation to the top of the chimney is \( 45^\circ \).

2. **Setting Up the Diagram**:
- Let \( P \) be the point on the ground from where the angles are measured.
- Let \( Q \) be the base of the house, \( R \) be the top of the house, and \( S \) be the top of the chimney.
- The height of the house \( QR = h \) and the height of the chimney \( RS = H \) (which we need to find).

3. **Using Trigonometric Ratios**:
- For the angle of elevation to the top of the chimney (\( S \)):
\[ \tan(45^\circ) = \frac{h + H}{d} \]
Since \( \tan(45^\circ) = 1 \):
\[ h + H = d \quad \text{(1)} \]

- For the angle of elevation to the top of the house (\( R \)):
\[ \tan(x^\circ) = \frac{h}{d} \]
This gives us:
\[ d = \frac{h}{\tan(x)} \quad \text{(2)} \]

4. **Equating the Two Expressions for \( d \)**:
- From equations (1) and (2), we can equate the two expressions for \( d \):
\[ h + H = \frac{h}{\tan(x)} \]

5. **Solving for \( H \)**:
- Rearranging the equation:
\[ H = \frac{h}{\tan(x)} - h \]
- Factoring out \( h \):
\[ H = h \left(\frac{1}{\tan(x)} - 1\right) \]
- Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \):
\[ H = h \left(\frac{\cos(x) - \sin(x)}{\sin(x)}\right) \]

6. **Final Expression for the Height of the Chimney**:
- Thus, the height of the chimney \( H \) is given by:
\[ H = h \left(\cot(x) - 1\right) \]

### Summary:
The height of the chimney can be expressed as:
\[ H = h \left(\cot(x) - 1\right) \]