Expand cos (A + B + C). Hence prove that cos A cos B cos C = sin A sin B cos C + sin B sin C cos A + sin C sin A cos B, if A + B + C = \(\frac{\pi}{2}\)

What must be the radius of a circular running path, around which an athlete must run 5 times in order to describe 1 km?

In a circular of diameter 40 cm, a chord is of length 20 cm. FInd the length of the minor arc of the chord?

If in two Circles, arcs of the same length subtend angles 60⁰ and 75⁰ at the center, find the ratio of their radii

Prove that sin 75^o - sin 15^o = cos 105^o + cos 15^o

Prove that sin⁸x - cos⁸x = (sin²x - cos²x)(1 - 2sin²x cos²x)

Prove that 4 cos 12° cos 48° cos 72° = cos 36°

Prove that 1 + cos 2x + cos 4x + cos 6x = 4 cos x cos 2x cos 3x.

If \(\sec x=\sqrt2\) and \(\frac{3\pi}{2}\) Find the value of \(\frac{1+\tan x+cosec x}{1+\cot x-cosec x}\)

Prove that 2\(sin^{ 2 }\frac { 3\pi }{ 4 } +2cos^{ 2 }\frac { \pi }{ 4 } +2sec^{ 2 }\frac { \pi }{ 3 } \) = 10