Question

Prove that tan A /1−cotA + cot A/ 1−tanA = 1 + tan A + cot A don't spam.

Answer 1

Answer:

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Answer 2

Answer:

How can I prove, tan A/1-cot A + cot A/1-tan A = 1+ tan A + cot A?

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L.H.S [math]= \displaystyle\frac{\text{tan}A}{1-\text{cot}A} + \displaystyle\frac{\text{cot}A}{1-\text{tan}A}[/math]

[math]= \displaystyle\frac{\text{tan}A}{1-\frac{1}{\text{tan}A}} + \displaystyle\frac{\frac{1}{\text{tan}A}}{1-\text{tan}A}[/math]

[math]= \displaystyle\frac{\text{tan}^2A}{\text{tan}A-1} - \displaystyle\frac{\frac{1}{\text{tan}A}}{\text{tan}A-1}[/math]

[math]=\displaystyle\frac{1}{\text{tan}A-1}\displaystyle\frac{\text{tan}^3A–1}{\text{tan}A}[/math]

[math]=\displaystyle\frac{1}{\text{tan}A-1}(\text{tan}A-1)\displaystyle\frac{\text{tan}^2A+\text{tan}A+1}{\text{tan}A}[/math]

[math]= \displaystyle\frac{\text{tan}^2A+\text{tan}A+1}{\text{tan}A}[/math]

[math]= \text{tan}A + 1 + \displaystyle\frac{1}{\text{tan}A}[/math]

[math]= \text{tan}A + 1 + \text{cot}A[/math]

[math]= [/math]R. H. S