Question 958763
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how do you find the exact value of the expression
cos(sin^-1 1/3 - tan^-1 1/2)
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        The solution in the post by @lwsshar3 is incorrect due to arithmetic error.

        I came to bring a correct solution.



<pre>
So, they want you find cos(arcsin(1/3) - arctan(1/2)).


An average school student will be shocked, I think.


Therefore, let's move forward accurately.


Let x = arcsin(1/3) and y = arctan(1/2).

So, sin(x) = 1/3 and x is in QI;

    tan(y) = 1/2 and y is in QI, too.


Then cos(x) = {{{sqrt(1-sin^2(x))}}} = {{{sqrt(1-(1/3)^2)}}} = {{{sqrt(1-1/9)}}} = {{{sqrt(8/9)}}} = {{{sqrt(8)/3}}}.


For 'y', tan(y) = 1/2  is the same as to say  sin(y) = {{{1/sqrt(5)}}},  cos(y) = {{{2/sqrt(5)}}}.


Therefore,

      cos(arcsin(1/3)-acrtan(1/2)) = cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y) = 

    = {{{(sqrt(8)/3)*(2/sqrt(5))}}} + {{{(1/3)*(1/sqrt(5))}}} = {{{(2*sqrt(8))/(3*sqrt(5))}}} + {{{1/(3*sqrt(5))}}} = {{{(2*sqrt(8)+1)/(3*sqrt(5))}}} = {{{(4*sqrt(2)+1)/(3*sqrt(5))}}} = {{{(4*sqrt(10)+sqrt(5))/15}}}.


<U>ANSWER</U>.  cos(arcsin(1/3) - arctan(1/2)) = {{{(4*sqrt(10)+sqrt(5))/15}}}. 
</pre>

Solved correctly.