Question

If cosectheta =13/12, then evaluate2 sintheta - 3 costheta/4 sintheta - 9 costhata

Answer 1

cosecθ=13/12
∴, sinθ=1/cosecθ=1/(13/12)=12/13
∴, cosθ   
=√(1-sin²θ)     [∵, sin²θ+cos²θ=1]
=√{1-(12/13)²}
=√(1-144/169)
=√25/169
=5/13
∴, 2sinθ-3cosθ/4sinθ-9cosθ
={(2×12/13)-(3×5/13)}/{(4×12/13)-(9×5/13)}
=(24/13-15/13)/(48/13-45/13)
=(9/13)/(3/13)
=9/13×13/3
=3 Ans.

Answer 2

Cosec θ = 13/12
so  Sinθ = 12/13     =>  Cos²θ  = 1 - sin² θ = 1 - 144/169 = 25/169
so  cosθ = 5/13

LHS = (2 Sin θ - 3 cosθ) / (4 sin θ - 9 cosθ)
        = (2 * 12 - 3 * 5)  / (4 * 12 - 9 * 5)         as 13 in denominator cancels out
        = 9/3 = 3