Question

Rewrite the expression so that there is no denominator:
(tan2x)/(cscx+1)

Answer 1

sin^2(x) + cos^2(x) = 1

cosec(x) = 1/sin(x), so divide by sin^2(x):

1 + cot^2(x) = cosec^2(x)
cot^2(x) = cosec^2(x) - 1
= (cosec(x) - 1)(cosec(x) + 1).

If we multiply top and bottom fraction by (cosec(x) + 1)

(cosec(x) + 1) tan^2(x) / (cosec(x) - 1)(cosec(x) + 1)
= (cosec(x) + 1) tan^2(x) / cot^2(x).

Now tan(x)
=1/cot(x) so tan^2(x)
=1/cot^2(x)

(cosec(x) + 1) tan^2(x) tan^2(x
= (cosec(x) + 1) tan^4(x) .
(cosec(x) + 1) tan^4(x).

Answer 2

`(tan2x)/(cscx+1)* (cscx-1)/(cscx - 1) = \\ (tan2x(cscx-1))/(csc^(2) x-1)`
csc² x - 1 = 1/sin²x - 1 = (1 - sin² x ) / sin² x = cos² x / sin² x = cot²x
... =
`(tan2x(cscx-1))/(cot ^(2) x)`=
= tan 2 x ( csc x - 1 ) tan² x