How can you rewrite this trigonometric expression as an algebraic expression?

Question

#cos(sin^-1(u)-cos^-1(v)#

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Answer 1 · 2018-03-09T07:44:28

#cos(sin^(-1)u-cos^(-1)v)=vsqrt(1-u^2)+usqrt(1-v^2)#

Let #sin^(-1)u=A# and #cos^(-1)v=B#

then #sinA=u#, which means #cosA=sqrt(1-u^2)#

and #cosB=v#, which means #sinB=sqrt(1-v^2)#

Hence, #cos(sin^(-1)u-cos^(-1)v)#

= #cos(A-B)#

= #cosAcosB+sinAsinB#

= #sqrt(1-u^2)xxv+uxxsqrt(1-v^2)#

= #vsqrt(1-u^2)+usqrt(1-v^2)#