Question #857b8

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Answer 1 · 2018-02-11T14:57:32

Kindly refer to a Proof in the Explanation.

$\frac{tan x}{1 - cot x} + \frac{cot x}{1 - tan x}$ $tanx/(1-cotx)+cotx/(1-tanx)$ ,

$= \frac{tan x}{1 - \frac{1}{tan x}} + \frac{\frac{1}{tan x}}{1 - tan x}$ $=tanx/(1-1/tanx)+(1/tanx)/(1-tanx)$ ,

$= \frac{{tan}^{2} x}{tan x - 1} - \frac{1}{tan x (tan x - 1)}$ $=tan^2x/(tanx-1)-1/{tanx(tanx-1)}$ ,

$= \frac{{tan}^{3} x - 1}{tan x (tan x - 1)}$ $=(tan^3x-1)/{tanx(tanx-1)}$ ,

$={cancel((tanx-1))(tan^2x+tanx+1)}/{tanxcancel((tanx-1))}$ ,

$=tan^2x/tanx+tanx/tanx+1/tanx$ ,

$=tanx+1+1/tanx$ ,

$=1+{sinx/cosx+cosx/sinx}$ ,

$=1+(sin^2x+cos^2x)/(sinxcosx)$ ,

$=1+1/(sinxcosx)$ ,

$=1+1/cosx*1/sinx$ ,

$=1+secxcscx,$ as desired!