1.

IF secthita+ tanthita =pThen find the value of cosecthita

Answer»
Sec2θ - tan2θ = 1(secθ+tanθ)(secθ-tanθ)=1or, secθ-tanθ=1/p ----------------(2)Adding (1) and (2) we get,2secθ=p+1/por, secθ=(p²+1)/2p∴, cosθ=1/secθ=2p/(p²+1)∴, sinθ=√(1-cos²θ)=√[1-{2p/(p²+1)}²]=√[1-4p²/(p²+1)²]=√[{(p²+1)²-4p²}/(p²+1)²]=√[(p⁴+2p²+1-4p²)/(p²+1)²]=√(p⁴-2p²+1)/(p²+1)=√(p²-1)²/(p²+1)=(p²-1)/(p²+1)∴, cosecθ=1/sinθ=1/[(p²-1)/(p²+1)]=(p²+1)/(p²-1)\xa0
Another method:Sec2θ - tan2θ = 1(secθ + tanθ)( secθ - tanθ) = 1p( secθ - tanθ) = 1( secθ - tanθ) = 1/p and, ......-(1)( secθ + tanθ) = p ..... - (2)(1) + (2),secθ = 1/2 (p + 1/p) ,(2) - (1)tanθ = 1/2 (p-1/p)secθ / tanθ = cosecθHence,cosecθ = p +1/p / p-1/p = p2\xa0+1 / p2\xa0-1


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