Trigonometry

We are given that: We know that: So, the equation becomes: This implies: Now, using the identity for the tangent of complementary angles: Thus, we have: Hence, we have proved that .

Step 1: Simplify each trigonometric expression.
We are asked to evaluate the expression: We know the following values:
- , so
- , so
-

Step 2: Substitute the values into the expression.
Substitute the values into the given expression:
Step 3: Simplify the expression.
To simplify, first find a common denominator: Thus, the value of the expression is:

Step 1: Express and using .
We are given . We know that: Thus, and for some constant . Also, since , we substitute:
Step 2: Substitute values of and into the expression.
Now that we know , we substitute and into the expression:
Step 3: Simplify the expression.
Use the difference of squares formula to simplify both the numerator and the denominator: Thus, the value of the expression is:

Step 1: Analyze the right triangle.
In this problem, we can form a right-angled triangle where the tower's height (AC) is the opposite side to the angle of elevation, and the distance from point B to the foot of the tower (AB) is the adjacent side. The angle of elevation is given as .
Step 2: Apply trigonometric ratios.
We will use the tangent of the angle of elevation: Given that and AC = 20m, we substitute the values into the equation:
Step 3: Solve for AB.
Since , we have: Solving for AB:
Step 4: Conclusion.
Thus, the distance from point B to the foot of the tower is: