If sec \(\theta\) + tan \(\theta\) = p, obtain the values of sec \(\theta\), tan \(\theta\) and sin \(\theta\) in terms of p

Eliminate \(\theta\) from the equation a sec \(\theta\) - c tan \(\theta\) = b and b sec \(\theta\)  + d tan \(\theta\) = C

Show that \(cot(A+{ 15 }^{ 0 })-tan(A-{ 15 }^{ 0 })=\frac { 4cos2A }{ 1+2sin2A } \)

If A + B + C = 180⁰, prove that \(tan\frac { A }{ 2 } tan\frac { B }{ 2 } +tan\frac { B }{ 2 } tan\frac { C }{ 2 } +tan\frac { C }{ 2 } tan\frac { A }{ 2 } =1\)

Solve the following equations sin \(\theta\) + sin 3\(\theta\) + sin5\(\theta\) = 0

Two soldiers A and B in two different underground bunkers on a straight road, spot an intruder at the top of a hill. The angle of elevation of the intruder from A and B to the ground level in the eastern direction are 30⁰ and 45⁰ respectively. If A and B stand 5 km apart, find the distance of the intruder from B

Two Navy helicopters A and B are flying over the Bay of Bengal at same altitude from the sea level to search a missing boat. Pilots of both the helicopters sight the boat at the same time while they are apart 10 km from each other. If the distance of the boat from A is 6 km and if the line segment AB subtends 60⁰ at the boat, find the distance of the boat from B.

If \(y=\frac{2\ sin\alpha}{1+cos\alpha+sin\alpha}\) then prove that \(\frac{1-cos\alpha+sin\alpha}{1+sin\alpha}=y\).

The Government plans to have a circular zoological park of diameter 8 km. A separate area in the form of a segment formed by a chord of length 4 km is to be allotted exclusively for a veterinary hospital in the park. Find the area of the segment to be allotted for the veterinary hospital.

Suppose that there are two cell phone towers within range of a cell phone. The two towers are located at 6 km apart along a straight highway, running east to west and the cell phone is north of the highway. The signal is 5 km from the first tower and \(\sqrt 31\)km from the second tower. Determine the position of the cell phone north and east of the first tower and how far it is from the highway.