Question 1189490
a) write {{{log(x^3 *sqrt((x+1))/(x-2)^2)}}}, {{{x>2}}} as a sum and difference logarithms.
Express all powers as factors.


 {{{log(x^3* sqrt((x+1))/(x-2)^2)}}}


= {{{log(x^3)+log( sqrt((x+1))/(x-2)^2)}}}


= {{{3log(x)+log( sqrt(x+1))-log((x-2)^2)}}}


= {{{3log(x)+log((x+1)^(1/2))-2log((x-2))}}}


= {{{3log(x)+(1/2)log((x+1))-2log((x-2))}}}


True for all {{{x}}}.
Verify solution {{{x>2}}}:  The solution is {{{x>2}}}


b) Find the exact value of the composite function {{{cos(sin^-1(-1/3))}}}

{{{cos(sin^-1(-1/3))}}}

let {{{sin^-1(-1/3)=theta}}} => {{{sin(theta)=-1/3}}}

This means that we are now looking for {{{cos(theta)}}}.

Next, use the identity : 

{{{cos^2(theta)+sin^2(theta)=1}}}

{{{cos(theta)=sqrt(1-sin^2(theta))}}}

{{{cos(theta)=sqrt(1-(-1/3)^2)}}}

{{{cos(theta)=sqrt(8/9)}}}

{{{cos(theta)=(2sqrt(2))/3}}}