If tanθ +1/tanθ=2,then show that tan²θ +1/tan²θ= 2 ,Prove it
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Answered by
5
Let
![\tan( \theta) = x \tan( \theta) = x](https://tex.z-dn.net/?f=+%5Ctan%28+%5Ctheta%29++%3D+x)
Then, We have
![x + \frac{1}{x} = 2 x + \frac{1}{x} = 2](https://tex.z-dn.net/?f=x+%2B++%5Cfrac%7B1%7D%7Bx%7D++%3D+2+)
Squaring both sides,
![{(x + \frac{1}{x} )}^{2} = {2}^{2} \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = 4 \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 4 \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 4 - 2 = 2 {(x + \frac{1}{x} )}^{2} = {2}^{2} \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 \times x \times \frac{1}{x} = 4 \\ = > {x}^{2} + \frac{1}{ {x}^{2} } + 2 = 4 \\ = > {x}^{2} + \frac{1}{ {x}^{2} } = 4 - 2 = 2](https://tex.z-dn.net/?f=+%7B%28x+%2B++%5Cfrac%7B1%7D%7Bx%7D+%29%7D%5E%7B2%7D++%3D++%7B2%7D%5E%7B2%7D+%5C%5C++%3D++%26gt%3B++%7Bx%7D%5E%7B2%7D+++%2B++%5Cfrac%7B1%7D%7B+%7Bx%7D%5E%7B2%7D+%7D++%2B+2+%5Ctimes+x+%5Ctimes++%5Cfrac%7B1%7D%7Bx%7D++%3D+4+%5C%5C++%3D++%26gt%3B++%7Bx%7D%5E%7B2%7D++%2B++%5Cfrac%7B1%7D%7B+%7Bx%7D%5E%7B2%7D+%7D++%2B+2+%3D+4+%5C%5C++%3D++%26gt%3B++%7Bx%7D%5E%7B2%7D++%2B++%5Cfrac%7B1%7D%7B+%7Bx%7D%5E%7B2%7D+%7D++%3D+4+-+2+%3D+2+)
Hence, we proved
![{tan}^{2} ( \theta) + \frac{1}{ {tan}^{2} ( \theta)} = 2 {tan}^{2} ( \theta) + \frac{1}{ {tan}^{2} ( \theta)} = 2](https://tex.z-dn.net/?f=+%7Btan%7D%5E%7B2%7D+%28+%5Ctheta%29+%2B++%5Cfrac%7B1%7D%7B+%7Btan%7D%5E%7B2%7D+%28+%5Ctheta%29%7D++%3D+2)
when
![\tan( \theta) + \frac{1}{ \tan( \theta) } = 2 \tan( \theta) + \frac{1}{ \tan( \theta) } = 2](https://tex.z-dn.net/?f=+%5Ctan%28+%5Ctheta%29+%2B++%5Cfrac%7B1%7D%7B+%5Ctan%28+%5Ctheta%29+%7D++%3D+2)
Then, We have
Squaring both sides,
Hence, we proved
when
Answered by
4
In the attachment I have answered this problem.
We use the following algebraic
identity to expand the given
expression .
(a+b) ^2 = a^2 + b^2 + 2ab
See the attachment for detailed solution.
We use the following algebraic
identity to expand the given
expression .
(a+b) ^2 = a^2 + b^2 + 2ab
See the attachment for detailed solution.
Attachments:
![](https://hi-static.z-dn.net/files/de6/020ec24ef0af264f627b83f232451bcb.jpg)
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