Write the trig function as an algebraic function cos(cos^-1 u + sin^-1 v)

Question

The textbook answer is (u√1-v² ) - (v√1- u²)

Answer 1

If cos^-1 u = φ then cos φ = u . But [sin φ]² +[ cosφ ]² = 1 ⇒ [sin φ]² = 1 - [ cosφ ]²

Then sin φ = √ {1 - [ cosφ ]²} and since cos φ = u Then sin φ = √( 1- u²).

In an exact similar manner

If sin^-1 v = θ then sin θ= v ⇒ cosθ = √( 1- v²)

Then cos [ cos^-1 u + sin^-1 v ] = cos [φ +θ] = cos φ cos θ - sin φ sinθ = u √( 1- v²) - v √( 1- u²) .