secx * cosecx - cotx = tanx

We know that:

secx = 1/cosx

csecx = 1/sinx

cot = cosx/sinx

tanx = sinx/cosx

Now substitute:

sec x cosec x - cot x = tan x \

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

1/sinx*cosx - cosx/sinx = sinx/cosx

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

But...

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secx * cosecx - cotx = tanx

We know that:

secx = 1/cosx

csecx = 1/sinx

cot = cosx/sinx

tanx = sinx/cosx

Now substitute:

sec x cosec x - cot x = tan x \

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

1/sinx*cosx - cosx/sinx = sinx/cosx

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

But we know that: sin^2 x + cos^2x = 1

==> 1-cos^2x = sin^2 x

==> sin^2 x /sin*cos = sin/cos

==> sinx /cosx = sinx/cos

==> tanx = tan x