Question

tanx/1-cotx + cotx/1-tanx = 1+secx cosecx

Answer 1

$\frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan x}=1+\sec x \csc x$ Proved.

Step-by-step explanation:

Consider the provided information.

We need to prove: $\frac{\tan x}{1-\cot x}+\frac{\cot x}{1-\tan x}=1+\sec x \csc x$

Consider the LHS

$\frac{\frac{\sin x}{\cos x}}{1-\frac{\cos x}{\sin x}}+\frac{\frac{\cos x}{\sin x}}{1-\frac{\sin x}{\cos x}}$

$\frac{\frac{\sin x}{\cos x}}{\frac{\sin x-\cos x}{\sin x}}+\frac{\frac{\cos x}{\sin x}}{\frac{\cos x-\sin x}{\cos x}}$

$\frac{\sin ^2x}{\cos x(\sin x-\cos x)}-\frac{\cos^2 x}{\sin x(\sin x-\cos x)}$

$\frac{\sin ^3x-\cos^3x}{\cos x\sin x(\sin x-\cos x)}$

$\frac{(\sin x-\cos x)(\sin^2x+\cos^2x+\sin x\cos x)}{\cos x\sin x(\sin x-\cos x)}$

$\frac{\sin^2x+\cos^2x+\sin x\cos x}{\cos x\sin x}$

$\frac{1+\sin x\cos x}{\cos x\sin x}$

$\frac{1}{\cos x\sin x}+\frac{\sin x\cos x}{\cos x\sin x}$

$\csc x\sec x+1$

$1+\sec x\csc x$

Hence, proved

#Learn more

Prove the trigonometric identity 1+ cot square A= cosec square A

https://brainly.in/question/2259148

Answer 2

1- Cotre

tanne + Cot x = 1 +tann + cotx = 14 Seck lose 1-tanne

LHS

(tan x)/(1 - cot x) + (cot x)/(1 - 4einx)

=((sin x)/(Cosx))/(1 - (sin x)/(sin x)) + ((cos x)/(sin x))/(1 - (sin x)/(cos x))

=((sin x)/(cos x))/((sin x - cos x)/(s!mc)) + ((cos x)/(sin x))/((cos x - sin x)/(cos x)) =(sin^2 x)/(cos x * (simx - cos x))

Sinx (sinn-casm)

=(li * n ^ 3 * x - cos^3 x)/(cos x * sin x * (sin x - asx))

=((sin x - cos x)(sin^2 x + cos^2 x + simc * cos x))/(cos x * sin x * (sin x - cos x))

=(sin^2 x + cos^2 x + sin x * cos x)/(cos xlinx)

=(1 + sin x * cos n)/(cos x * sin x)

=1/(cos x * sin x) + (sin x * cos x)/(cos x * sin x)

=sec(x) * cos ecx + 1

=1 + sec(x)

RHS

=1 + seex caseen

LHC = 2HS veuifred

Caseen