Simplifying trig identities

Question

Simplify using trig identities

csc(x)-cot(x)/sec(x)-1

Answer 1

cscx - cotx/secx -1

= 1/sinx -(cosx/sinx)/(1/cosx) -1

= 1/sinx = (cosx/sinx)(cosx)-1

=1/sinx- cos^2x/sinx -1

= (1-cos^2x)/sinx-1

= sin^2x/sinx -1

= sinx -1

Unless you meant

cscx - cotx/(secx -1)

then

1/sinx - (cosx/sinx)/(1/cosx-1)

= 1/sinx - cosx/(sinxcosx- sinx)

= (1-cosx/(cox-1))

=(cosx-1- coxx)/(cosx-1)

=-1/(cosx-1)

= 1/(1-cosx)

= (1-cosx)^-1

Or did you mean

cscx - cot/sec^-1(x) ??

if so, then

1/sinx -(cosx/siinx)(cosx)

= 1/sinx- cos^2x/sinx

= (1-cos^2)/sinx

= sin^2x/sinx

= sinx

Or did you possibly mean

cscx - (cotx/secx)^-1 ??

if so, then

1/sinx - secx/cotx

= 1/sinx - (1/cosx)/(cosx/sinx)

= 1/sinx -(1/cosx)(sinx/cosx)

= 1/sinx -sinx/cos^2(x)

= cos^2(x)/sinxcos^2(x) -sin^2(x)/sinxcos^2(x)

=(cos^2(x)- sin^2(x))/sinxcos^2(x)

= cos2x/sinxcos^2(x)

Usual techique, if you don't see anything else, is convert everything into sines and cosines and hope for the best.