SOLUTION: cot(theta) = (2n^2+2mn)/(m^2+2mn) , then prove that cosec(theta)=(m^2+2mn+2n^2)/(m^2+2mn)

Question 938710: cot(theta) = (2n^2+2mn)/(m^2+2mn) , then prove that cosec(theta)=(m^2+2mn+2n^2)/(m^2+2mn)
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
cot(theta) = (2n^2+2mn)/(m^2+2mn) ,
then prove that
cosec(theta)=(m^2+2mn+2n^2)/(m^2+2mn)
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Since cot = y/x, y = 2n^2+2mn and x = m^2+2mn
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Similarly, csc = r/x
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r = sqrt[x^2+y^2] = sqrt[(m^2+2mn)^2 + (2n^2+2mn)^2]
---
= sqrt[(m(m+2n))^2 +(2n(n+m))^2]
---
= sqrt[(m^2(m^2+4mn+4n^2)) + (4n^2(n^2+2mn+m^2))]
====
= sqrt[m^4 + 4m^3n + 4m^2n^2 + 4n^4 + 8mn^3 + 4n^2m^2]
----
= sqrt[m^4 + 4m^3n + 8m^2n^2 + 8mn^3 + 4n^4]
---
= sqrt[m^2+2mn+2n^2]^2
----
= m^2+2mn+2n^2
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Therefore csc(theta) = r/x = (m^2+2mn+2n^2)/(m^2+2mn)
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Cheers,
Stan H.
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