Question

Statement -1: If `xin[-1//sqrt(3),1//sqrt(3)]` then `cot^(-1)((3x-x^(3))/(1-3x^(2)))=cos^(-1)((1-x^(2))/(1+x^(2)))rarrx=sqrt(25-10sqrt(5))/(5)` statement -2: `sin18^(@)=sqrt(5-1)/(4) and cos18^(@)=sqrt(10+2sqrt(5))/4 `
A. Statement-1 is is True, Statement-2 is true, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer 1

Clearly statement 2 is true
now `cot^(-1)(3x-x^(3))/(1-3x^(2))=cos^(-1)(1-x^(2))/(1+x^(2))`
`rarr (pi)/(2) tan^(-1)(3x-x^(3))/(1-3x^(2))=cos^(-1)(1-x^(2))/(1+x^(2))`
`rarr (pi)/(2) -3 tan^(-1)x=2 tan^(-1)x`
`rarr tan^(-1)x=(pi)/(10)`
`rarr x=tana(pi)/(10)=(sin 18^(@))/(cso 18^(@))=(sqrt(5)-1)/(sqrt(10+2sqrt(5))`
`rarr x=(sqrt(5)-1sqrt(10-2)sqrt(5))/(sqrt(100-20) rarr x=(sqrt(10-2)sqrt(5)(6-2sqrt(5))/(4sqrt(5))`
`rarr x =sqrt(80-32sqrt(5))/(4sqrt(5))=sqrt(5-2sqrt(5))/sqrt(5)=sqrt(25-10)sqrt(5)/(5)`
so statement -1 is true
Also statemnet -2 is correct explanation for statement -1