Proving Identities?

Question

prove the following identity:

A)

(1+cosx) / (1-cosx) = (1+secx) / (secx-1)

B)

(sinx+tanx) / (cosx+1) = tanx

Answer 1

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LHS

= (1 + cos(x)) / (1 - cos(x))

= ((1 + cos(x))/cos(x)) / ((1-cos(x))/cos(x))

= (sec(x) + 1) / (sec(x) - 1)

= (1 + sec(x)) / (sec(x) - 1)

= RHS

LHS

= (sin(x) + tan(x)) / (cos(x) + 1)

= (sin(x) + sin(x)/cos(x)) / (cos(x) + 1)

= sin(x)(1 + 1/cos(x)) / (cos(x) + 1)

= sin(x)cos(x)(1 + 1/cos(x)) / (cos(x)(cos(x) + 1))

= sin(x)(cos(x) + 1) / (cos(x)(cos(x) + 1))

= sin(x) / cos(x)

= tan(x)

= RHS