$$ABC$$ is a triangular park with $$AB=AC=100$$ $$m$$. A television tower stands at the midpoint of $$BC$$. The angles of elevetion of the top of the tower at $$A, B, C$$ are 45$$^ \circ $$, 60$$^ \circ $$, 60$$^ \circ $$, respectively. Find the height of the tower.

A sign -post in the form of an isosceles triangle $$ABC$$ is mounted on a pole of height $$h$$ fixed to the ground. The base $$BC$$ of the triangle is parallel to the ground. A man standing on the ground at a distance $$d$$ from the sign-post finds that the top vertex $$A$$ of the triangle subtends an angle $$\beta $$ and either of the other two vertices subtends the same angle $$\alpha $$ at his feet. Find the area of the triangle.

If in a triangle $$ABC$$, $$\cos A\cos B + \sin A\sin B\sin C = 1,$$ Show that $$a:b:c = 1:1:\sqrt 2 $$

A ladder rests against a wall at an angle $$\alpha $$ to the horizintal. Its foot is pulled away from the wall through a distance $$a$$, so that it slides $$a$$ distance $$b$$ down the wall making an angle $$\beta $$ with the horizontal. Show that $$a = b\tan {1 \over 2}\left( {\alpha + \beta } \right)$$