Prove that: tanx*sinx / (sec^2 x -1)  = cosx

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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We need to prove that:

(tanx*sinx) / (sec^2 x -1) = cosx

We will start from the left side and prove that it equals cosx.

First we will rewrite using the trigonometric identities.

We know that secx = 1/cosx

==> sec^2 x = 1/cos^2 x

==> (tanx*sinx)/(sec^2 x -1) = (tanx*sinx)/(1/cos^2 x   -1)

                                            = (tanx*sinx)/[ (1-cos^2 x)/ cos^2x]

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

Also, we know that 1-cos^2 x = sin^2

==> cos^2x(tanx*sinx) / sin^2 x

Now we know that tanx = sinx/cosx

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

==> (cos^2 x * sin^2 x) / cosx * sin^2 x

Now we will reduce similar.

==> cosx ...... q.e.d

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