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We start from the equation
cot A + cosec A = 3 can be written as
cos A/sin A + 1/sin A = 3, or
cos A + 1 = 3 sin A, or
cos^2 (A/2) - sin^2 (A/2) + cos^2 (A/2) + sin^2 (A/2) = 3*2 sin (A/2)cos (A/2), or
2 cos^2 (A/2) =3*2 sin (A/2)cos (A/2), or
cos (A/2) = 3sin (A/2), or
tan (A/2) = 1/3
tan A = 2tan (A/2)/[1 - tan^2 A]
= 2*(1/3)/(1 - 1/9)
=(2/3)/(8/9), or
=2*9/3*8
= 3/4
So Pythagoras comes in here, height of right-angled triangle is 3, its base is 4, hence the hypotenuse is 5, therefore
cos A = 4/5 and sin A = 3/5
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