tanA cotA
---------- + -------------- = 1+cosecA×secA
1-cotA 1-tan A
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Formula:
(1)![( x^{3} - y^{3}) = (x - y) ( x^{2} + (x)(y) + y^{2}) ( x^{3} - y^{3}) = (x - y) ( x^{2} + (x)(y) + y^{2})](https://tex.z-dn.net/?f=%28+x%5E%7B3%7D+-++y%5E%7B3%7D%29+%3D+%28x+-+y%29+%28+x%5E%7B2%7D+%2B+%28x%29%28y%29+%2B++y%5E%7B2%7D%29)
(2)![sin^{2}(\theta) + cos^{2}(\theta) = 1 sin^{2}(\theta) + cos^{2}(\theta) = 1](https://tex.z-dn.net/?f=sin%5E%7B2%7D%28%5Ctheta%29+%2B+cos%5E%7B2%7D%28%5Ctheta%29+%3D+1)
(3)![tan(\theta) = \frac{sin(\theta)}{cos(\theta)} tan(\theta) = \frac{sin(\theta)}{cos(\theta)}](https://tex.z-dn.net/?f=tan%28%5Ctheta%29+%3D++%5Cfrac%7Bsin%28%5Ctheta%29%7D%7Bcos%28%5Ctheta%29%7D)
(4)![cot(\theta) = \frac{cos(\theta)}{sin(\theta)} cot(\theta) = \frac{cos(\theta)}{sin(\theta)}](https://tex.z-dn.net/?f=cot%28%5Ctheta%29+%3D++%5Cfrac%7Bcos%28%5Ctheta%29%7D%7Bsin%28%5Ctheta%29%7D)
(5)![sec(\theta) = \frac{1}{cos(\theta)} sec(\theta) = \frac{1}{cos(\theta)}](https://tex.z-dn.net/?f=sec%28%5Ctheta%29+%3D+%5Cfrac%7B1%7D%7Bcos%28%5Ctheta%29%7D)
(6)![cosec(\theta) = \frac{1}{sin(\theta)} cosec(\theta) = \frac{1}{sin(\theta)}](https://tex.z-dn.net/?f=cosec%28%5Ctheta%29+%3D+%5Cfrac%7B1%7D%7Bsin%28%5Ctheta%29%7D)
![\frac{tan(A)}{1 - cot(A)} = \frac{ \frac{sin(A)}{cos(A)} }{1 - \frac{cos(A)}{sin(A)} } = \frac{sin^{2}(A)}{cos(A)(sin(A) - cos(A))} \frac{tan(A)}{1 - cot(A)} = \frac{ \frac{sin(A)}{cos(A)} }{1 - \frac{cos(A)}{sin(A)} } = \frac{sin^{2}(A)}{cos(A)(sin(A) - cos(A))}](https://tex.z-dn.net/?f=%5Cfrac%7Btan%28A%29%7D%7B1+-+cot%28A%29%7D+%3D++%5Cfrac%7B+%5Cfrac%7Bsin%28A%29%7D%7Bcos%28A%29%7D+%7D%7B1+-++%5Cfrac%7Bcos%28A%29%7D%7Bsin%28A%29%7D+%7D+%3D++%5Cfrac%7Bsin%5E%7B2%7D%28A%29%7D%7Bcos%28A%29%28sin%28A%29+-+cos%28A%29%29%7D)
![\frac{cot(A)}{1 - tan(A)} = \frac{ \frac{cos(A)}{sin(A)} }{1 - \frac{sin(A)}{cos(A)} } = \frac{cos^{2}(A)}{sin(A)(cos(A) - sin(A))} \frac{cot(A)}{1 - tan(A)} = \frac{ \frac{cos(A)}{sin(A)} }{1 - \frac{sin(A)}{cos(A)} } = \frac{cos^{2}(A)}{sin(A)(cos(A) - sin(A))}](https://tex.z-dn.net/?f=%5Cfrac%7Bcot%28A%29%7D%7B1+-+tan%28A%29%7D+%3D++%5Cfrac%7B+%5Cfrac%7Bcos%28A%29%7D%7Bsin%28A%29%7D+%7D%7B1+-++%5Cfrac%7Bsin%28A%29%7D%7Bcos%28A%29%7D+%7D+%3D++%5Cfrac%7Bcos%5E%7B2%7D%28A%29%7D%7Bsin%28A%29%28cos%28A%29+-+sin%28A%29%29%7D)
Thus,
![\frac{tan(A)}{1 - cot(A)} +\frac{cot(A)}{1 - tan(A)}\\ \\ =\frac{sin^{2}(A)}{cos(A)(sin(A) - cos(A))} +\frac{cos^{2}(A)}{sin(A)(cos(A) - sin(A))}\\ \\ =\frac{sin^{2}(A)}{cos(A)(sin(A) - cos(A))} - \frac{cos^{2}(A)}{sin(A)(sin(A) - cos(A))}\\ \\ =\frac{sin^{3}(A)}{sin(A)cos(A)(sin(A) - cos(A))} - \frac{cos^{3}(A)}{sin(A)cos(A)(sin(A) - cos(A))} \frac{tan(A)}{1 - cot(A)} +\frac{cot(A)}{1 - tan(A)}\\ \\ =\frac{sin^{2}(A)}{cos(A)(sin(A) - cos(A))} +\frac{cos^{2}(A)}{sin(A)(cos(A) - sin(A))}\\ \\ =\frac{sin^{2}(A)}{cos(A)(sin(A) - cos(A))} - \frac{cos^{2}(A)}{sin(A)(sin(A) - cos(A))}\\ \\ =\frac{sin^{3}(A)}{sin(A)cos(A)(sin(A) - cos(A))} - \frac{cos^{3}(A)}{sin(A)cos(A)(sin(A) - cos(A))}](https://tex.z-dn.net/?f=%5Cfrac%7Btan%28A%29%7D%7B1+-+cot%28A%29%7D+%2B%5Cfrac%7Bcot%28A%29%7D%7B1+-+tan%28A%29%7D%5C%5C+%5C%5C+%3D%5Cfrac%7Bsin%5E%7B2%7D%28A%29%7D%7Bcos%28A%29%28sin%28A%29+-+cos%28A%29%29%7D+%2B%5Cfrac%7Bcos%5E%7B2%7D%28A%29%7D%7Bsin%28A%29%28cos%28A%29+-+sin%28A%29%29%7D%5C%5C+%5C%5C+%3D%5Cfrac%7Bsin%5E%7B2%7D%28A%29%7D%7Bcos%28A%29%28sin%28A%29+-+cos%28A%29%29%7D+-+%5Cfrac%7Bcos%5E%7B2%7D%28A%29%7D%7Bsin%28A%29%28sin%28A%29+-+cos%28A%29%29%7D%5C%5C+%5C%5C+%3D%5Cfrac%7Bsin%5E%7B3%7D%28A%29%7D%7Bsin%28A%29cos%28A%29%28sin%28A%29+-+cos%28A%29%29%7D+-+%5Cfrac%7Bcos%5E%7B3%7D%28A%29%7D%7Bsin%28A%29cos%28A%29%28sin%28A%29+-+cos%28A%29%29%7D)
(1)
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Thus,
Answered by
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[tex] \frac{tan\ A}{1-\frac{1}{tan\ A}} + \frac{1/tan\ A}{1 - tan\ A} \\ \\
\frac{tan^{2}\ A}{tan\ A -1} + \frac{1}{tan\ A\ (1 - tan\ A)} \\ \\
\frac{- tan^{3}\ A +1}{tan\ A\ (1 - tan\ A)} \\ \\ \frac{(1-tan\ A)(1+tan\ A + tan^{2}\ A)}{tan\ A (1-tan\ A)} \\ \\
1 + cot\ A + tan\ A \\ \\ 1 + \frac{cos^{2}\ A + sin^{2}\ A}{cos\ A\ sin\ A} \\ \\
Hence\ the\ answer\ on\ the\ RHS \\
[/tex]
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