Question

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

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Answer 1

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)