What does $cos(arctan(3))+sin(arctan(4))$ equal?

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Answer 1 · 2018-03-04T04:02:18

$cos(arctan(3))+sin(arctan(4))=1/sqrt(10)+4/sqrt(17)$

Let $tan^-1(3)=x$

then $rarrtanx=3$

$rarrsecx=sqrt(1+tan^2x)=sqrt(1+3^2)=sqrt(10)$

$rarrcosx=1/sqrt(10)$

$rarrx=cos^(-1)(1/sqrt(10))=tan^(-1)(3)$

Also, let $tan^(-1)(4)=y$

then $rarrtany=4$

$rarrcoty=1/4$

$rarrcscy=sqrt(1+cot^2y)=sqrt(1+(1/4)^2)=sqrt(17)/4$

$rarrsiny=4/sqrt(17)$

$rarry=sin^(-1)(4/sqrt(17))=tan^(-1)4$

Now, $rarrcos(tan^(-1)(3))+sin(tan^(-1)tan(4))$

$rarrcos(cos^-1(1/sqrt(10)))+sin(sin^(-1)(4/sqrt(17)))=1/sqrt(10)+4/sqrt(17)$

What does cos(arctan(3))+sin(arctan(4))cos(arctan(3))+sin(arctan(4)) equal?