Q. 304.4( 7 Votes )

If sec θ + tan θ

Answer :

Given sec θ + tan θ = p


But we know ,


substituting these values in the given equation, we get




Now we will square on both sides, we get




Now we know sin2θ + cos2θ = 1 cos2θ = 1 - sin2θ, substituting this in above equation, we get



Now expanding the numerator by applying the formula (a + b)2 = a2 + b2 + 2ab, we get




Now let x = sin θ, so the above equation becomes



1 + x2 + 2x = p2(1 - x2)


1 + x2 + 2x = p2 - p2x2


1 + x2 + 2x - p2 + p2x2 = 0


(1 + p2)x2 + 2x + (1 - p2) = 0


Comparing this with standard equation, i.e., Ax2 + Bx + C = 0, we get


A = (1 + p2), B = 2, C = (1 - p2)


So the value of x of a quadratic equation can be found by using the formula,



Now substituting the corresponding values, we get








So the two possibilities are,





But we had assumed x = sin θ, substituting back we get



And we know



Substituting the value of sin θ, we get




These are the required value of cosec θ.


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