Prove that sin² Acos² B + cos² Asin² B + cos² Acos² B + sin² Asin² B=1

if cos\(\theta \) + sin\(\theta \) =\(\sqrt { 2 } \) cos \(\theta \), then prove that cos\(\theta \) - sin\(\theta \) =\(\sqrt { 2 } \) sin\(\theta \) 

if cosec\(\theta \) + cot\(\theta \) = p, then prove that cos\(\theta \) = \(\frac { { p }^{ 2 }-1 }{ { p }^{ 2 }+1 } \)

if sin\(\theta \) + cos\(\theta \)  = \(\sqrt { 3 } \),then prove that tan\(\theta \) + cot\(\theta \) = 1

if \(\frac { cos\theta }{ 1+sin\theta } =\frac { 1 }{ a } \),then prove that \(\frac { { a }^{ 2 }-1 }{ a^{ 2 }+1 } \) = sin\(\theta \)

Two ships are sailing in the sea on either sides of a lighthouse as observed from the ships are \(30°\) and \(45°\) respectively. if the lighthouse is 200 m high, find the distance between the two ships. \(\left( \sqrt { 3 } =1.732 \right) \)

A kite is flying at a height of 75m above the ground, the string attached to the kite is temporarily tied to a point on the ground. The inclination of the string with the ground is \(60°\).find the length of the string ,assuming that there is no slack in the string.

From a point on the ground, the angles of elevation of the bottom and top of a tower fixed at the top of a 30m high building are \(45°\)and \(60°\) respectively. find the height of the tower. (\(\sqrt { 3 } =1.732\) )

A tv tower stands vertically on a bank of a canal. the tower is watched from a point on the other bank directly opposite to it. the angel of elevation of the top of the tower is 58°. from another point 20m away from this point on the line joining this  point of the tower, the angel of elevation of the top of the tower is 30°.find the height of the tower and the width of the canal.( tan58°=1.6003)

An Aeroplane sets of from G on bearing of 24° towards H, a point 250 km away, at H it changes  course and heads towards J deviates further by 55° and a distance of 180 km away.
How far is H to the north of G?,
\(\left( \begin{matrix} sin24°=0.4067\quad sin11°=0.1908 \\ cos24°=0.9135\quad cos11°=0.9816 \end{matrix} \right) \)