[cos(x)(sec(x) - cos(x))] / sin(x)
Change everything in terms of cos and sin:
= [cos(x)((1 / cos(x)) - cos(x))] / sin(x)
Distribute cos(x):
= [1 - cos^2(x)] /sin(x)
Trig identity:
= sin^2(x) / sin(x)
Multiply numerator and denominator by 1/sin(x):
= sin(x)
cosx(sec x - cosx)/sinx
=[cosx secx - cos^2x]/sinx
=[1-cos^2x]sinx
= sin^x/sinx = sinx
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[cos(x)(sec(x) - cos(x))] / sin(x)
Change everything in terms of cos and sin:
= [cos(x)((1 / cos(x)) - cos(x))] / sin(x)
Distribute cos(x):
= [1 - cos^2(x)] /sin(x)
Trig identity:
= sin^2(x) / sin(x)
Multiply numerator and denominator by 1/sin(x):
= sin(x)
cosx(sec x - cosx)/sinx
=[cosx secx - cos^2x]/sinx
=[1-cos^2x]sinx
= sin^x/sinx = sinx