Prove this Trig Identity: (sinx-1)(tanx + secx) = -cosx?

Hi,

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Multiply out:

(sin(x) - 1)(tan(x) + sec(x)) = sin(x)tan(x) - tan(x) + sin(x)sec(x) - sec(x)

Replace tan(x) by sin(x)/cos(x) and sec(x) by 1/cos(x):

sin(x)[sin(x)/cos(x)] - sin(x)/cos(x) + sin(x)/cos(x) - 1/cos(x)

The middle two terms cancel. Multiply out the first term and combine it with the last term

sin²(x)/cos(x) - 1/cos(x)

(sin²(x) - 1)/cos(x)

Use the identity sin²(x) = 1 - cos²(x):

(1 - cos²(x) - 1) / cos(x)

which simplifies to -cos(x)