Prove that: `tan^(-1)4/5+cos^(-1)(12)/(13)=cos^(-1)(33)/(65)`

Explanation

Let `cos ^(-1)((4)/(5))=x,cos^(-1)((12)/(13))=y and cos^(-1)((33)/(65))=z`.
Then `cosx=(4)/(5), "where " 0 lt x lt (pi)/(2)`
`cos y=(12)/(13), "where " 0 lt y lt (pi)/(2)`
and `cos z=(33)/(65),"where " 0 lt z lt (pi)/(2)`.
`therefore sin x gt 0 ` and `sinygt 0`.
Now , `sin x= sqrt(1-cos^(2)x)=sqrt(1-(16)/(25))=sqrt((9)/(25))=(3)/(5)`
and `sin y=sqrt(1-cos^2y)=sqrt(1-(144)/(169))=sqrt((25)/(169))=(5)/(13)`
We have to show that ,`x+y=z`
Now , `cos(x+y)=cosx cosy-sinxsiny=((4)/(5))((12)/(13))-((3)/(5))((5)/(13))=(48)/(65)-(15)/(65)=(33)/(65)`
`therefore cos (x+y)=cos z`
`therefore x+y=z`
`therefore cos^(-1)((4)/(5))+cos^(-1)((12)/(13))=cos^(-1)((33)/(65))`.