Question

Simplify the expression ?

Answer 1

The answer is cscx(cscx - 1) ⇒ the second answer

Explanation:

∵ csc²x = cot²x + 1

∴
`(cot^(2)x+1-1 )/(1+sinx)=(cot^(2)x )/(1+sinx)`

Multiply the fraction by its conjugate 1 - sinx (up and down)

∴
`(cot^(2)x )/(1+sinx)*(1-sinx)/(1-sinx)=(cot^(2)x(1-sinx) )/(1-sin^(2)x)`

∵ 1 - sin²x = cox²x

∴
`(cot^(2)x(1-sinx) )/(cos^(2)x )`

∵ cot²x = cos²x/sin²x

∴
`(1-sinx)/(sin^(x))=(1)/(sin^(2)x)-(sinx)/(sin^(2)x)=csc^(2)x-cscx`

Take cscx as a common factor

∴ cscx(cscx - 1)