If cosec θ – sin θ = b, sec θ – cos θ = a           sinθcos θ = ???

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justaguide | College Teacher | (Level 2) Distinguished Educator

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I'm going to use x instead of theta.

It is given that cosec x - sin x = b and sec x - cos x = a.

We need to find sin x * cos

sec x = 1/cos x and cosec x  = 1/ sin x

cosec x - sin x = b and sec x - cos x = a

=> 1/ sin x - sin x = b and 1/cos x - cos x = a

b = [1 - (sin x)^2]/sin x and a = [1 - (cos x)^2]/cos x

=> b = (cos x)^2 / sin x and a = (sin x)^2/ cos x

b *a = [(cos x)^2 / sin x]*[(sin x)^2/ cos x]

=> [(cos x)^2 *(sin x)^2 / sin x*cos x]

=> sin x * cos x

This gives us (sin x)*(cos x) = a*b

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