Can someone help me prove sinx/(secx+1)+sinx/(secx-1)=2cotx and (tanx+1)/(tanx-1)=(secx+cscx)/(secx-cscx)?

sinx/(secx+1) + sinx/(secx-1) = 2cotx

now:

starting from LHS group the two terms:

[sinx(secx-1) + sinx(secx +1)] / [sec^2(x) - 1] = 2cotx

[sinxsecx - sinx + sinxsecx +sinx]/ [sec^2(x) -1] = 2cotx

2sinxsecx/ [sec^2(x) -1] = 2cotx ..............................(1)

now sinxsecx = sinx/cosx = tanx

and sec^2(x) -1 = 1/cos^2(x) - 1 = [1-cos^2(x)/(cos^2(x))

So (1) becomes:

2sinx/cosx/ [ 1-cos^2(x)/ (cos^2(x)] = 2sinx/cosx/ (sin^2(x)/cos^2(x))

= 2tanx/ tan^2(x)

= 2/tanx

= 2cotx

Try starting from RHS, try to write secx and cscx in terms of tanx:

cscx = 1/sinx= 1/(cosxtanx)

whereas secx = 1/cosx= 1/(sinx/tanx) = tanx/sinx

So we get:

[(tanx/sinx + 1)/ cosxtanx) ]/ [ (tanx/sinx - 1)/ cosxtanx ]

Group together terms, now we can just add the two terms together BECAUSE sinx = cosxtanx

in other words, the above statement becomes:

[ tanx/cosxtanx + 1/cosxtanx ]/ [ tanx/cosxtanx - 1/cosxtanx ]

adding terms:

[(tanx +1)/cosxtanx]/[(tanx -1)/cosxtanx]

cancel cosxtanx:

(tanx +1)/ (tanx-1), which is the LHS of the statement that you needed to prove!

Multiply it out. You get: secx - sinxsecx + tanx -sinxtanx = a million/cosx -sinx/cosx + tanx - sin^2 (x) /cosx = a million/cosx - tanx + tanx - sin^2(x) /cosx = (a million - sin^2(x)) / cosx = cos^2(x) /cosx = cosx QED