Given that 2 sec?0 – tan²0 = p. show that cosec²theta= p-1/p-2,p≠2

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Step-by-step explanation:

2/cos^2(theta) - sin^2(theta)/cos^(theta) = p

(2 - sin^2(theta) ) / cos^2(theta) = p

cos^2(theta) = 1 - sin^2(theta)                    Relationship between sines and cosines

2 - sin^2(theta)/ (1 - sin^2(theta) ) = p        Everything is now in terms of sines

sin^2 (theta) = 1 / csc ^2 (theta)                 sin^(theta) = 1/csc(theta)

2 - 1/csc^2(theta)                                        Make Left over csc(theta)

       =  p

1 - 1/csc^2(theta)

2 csc^2(theta) - 1

csc^2(theta)

   = p         Cancel out denominators (csc^2(theta))

csc(theta)  - 1

csc^2(theta)

2 csc^2 (theta) - 1

     = p              Multiply both sides by csc^2(theta) - 1

csc^2(theta) - 1

2csc^2(theta) - 1 = p*csc^2(theta) - p   Collect csc^2(theta) on the left, p on the right.

csc^2(theta) (2 - p)  = 1 - p

csc^2(theta) = (1 - p)/(2 - p)