Solve for `x :` `cos(tan^(-1)x)=sin(cot^(-1)""3/4)`

Explanation

Let `cot^(-1)3/4=theta`. Then, `cottheta=3/4`
Draw a right triangle in which base = 3 units and perpendicular =4 units.
Then, hypotenuse `=sqrt(3^(2)+4^(2))=sqrt(25)=5` units
`therefore sintheta=4/5 rArr theta=sin^(-1)4/5`.
So, `sin(cot^(-1)3/4)=sin(sin^(-1)4/5)=4/5`.
Let `tan^(-1)x=phi`. Then, `tanphi=x/1`.
Draw a right triangle in which perpendicular =x and base =1.
Then, hypotenuse `=sqrt(1+x^(2))`.
`therefore cosphi=1/sqrt(1+x^(2))`
So, `cos(tan^(-1))=cosphi=1/sqrt(1+x^(2))`.
Thus, `1/sqrt(1+x^(2))=4/5 rArr 1/(1+x^(2))=16/25 rArr 16/25 rArr 161x^(2)=9`.
`rArr x^(2)=9/16 rArr x=+-34`.
Hence, `x=+-3/4`.