Prove the following: `tan^(-1)1/4+tan^(-1)2/9=1/2cos^(-1)3/5=1/2sin^(-1)4/5`

Explanation

We have,
`tan^(-1)1/4+tan^(-1)2/9=tan^(1){(1/4+/2/9)/(1-(1/4 xx 2/9))}=tan^(-1)1/2`
Now `2tan^(-1)x=cos^(-1)(1-x^(2))/(1+x^(2)) rArr tan^(-1)x=1/2cos^(-1)(1-x^(2))/(1+x^(2))`
`rArr tan^(-1)1/2=1/2cos^(-1)(1-1/4)/(1+1/4)=1/2cos^(-1)3/5`.
And, `2tan^(-1)=1/2cos^(-1))1-1/4)/(1+1/4)=1/2cos^(-1)3/5`.
And, `2tan^(-1)x=sin^(-1)(2x)/(1+x^(2))rArr tan^(-1)x=1/2sin^(-1)(2x)/(1+x^(2))`
`rArr tan^(-1)1/2=1/2sin^(-1)(2 xx 1/2)/(1+1/4)=1/2sin^(-1)4/5`.
Hence, `tan^(-1)1/4+tan^(-1)2/9=1/2cos^(-1)3/5=1/2sin^(-1)4/5`.