Question

IF sec A + tan A = p then, find the value of cosec A. SOLVED reply

Answer 1

secθ+tanθ=p ----------------------(1)
∵, sec²θ-tan²θ=1
or, (secθ+tanθ)(secθ-tanθ)=1
or, secθ-tanθ=1/p ----------------(2)
Adding (1) and (2) we get,
2secθ=p+1/p
or, 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) Ans.

Answer 2

Step-by-step explanation:

secθ+tanθ=p ......... (1)

∵sec2θ−tan2θ=1

or (secθ+tanθ)(secθ−tanθ)=1

or secθ−tanθ=1/p ....... (2)

Adding (1) & (2) we get

2secθ=p+1/p

or secθ=(p2+1)/2p

∴cosθ=1/secθ=2p/(p2+1)

∴sinθ=(1−cos2θ)

=[1−{2p/(p2+1)}2]=1−4p2/(p2+1)2=p4+2p2+1−4p2/(p2+1)2

=(p4−2p2