rewrite the following as nonfractional format 1a 1sin2x csc

rewrite the following as non-fractional format

1a. (1-sin^2x) / (csc - 1)

1b. sin^2y / (csc^2 - 1)

1c. csc(-x) / sec (-x)

Solution

1a. (1-sin^2x) / (csc - 1)

(1-sin^2x) /( 1-sinx)/sinx

Numerator (1-sin^2x) = (1+sinx)(1-sinx)

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

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

1b. sin^2y / (csc^2 - 1)

sin^2y/( 1/sin^2y -1)

sin^2y/ ( 1- sin^2y)/sin^2y

sin^2y*sin^2y/( 1-sin^2y)

=sin^4y/cos^2y

= tan^2y*sec^2y

1c. csc(-x) / sec (-x)

we know csc(-x) = -cscx and sec(-x) = secx

-cscx/secx

-1/sinx / 1/cosx

-cotx