Prove trigonometry

Question

Prove; (tanθ/(1-cotθ))+ (cotθ/(1-tanθ)) = 1+tanθ

Answer 1

The left side of equation is not equal to right side for all values of theta

so equation is not an identity.

Theta=0 is a problem value. right side=1,,,,left side=+/- infinity

[tan(x)/(1-cot(x))]+[cot(x)/(1-tan(x))]=1+2csc(2x)

2csc(2x)≠tan(x)

Answer 2

t/(1-c) +c/(1-t) = 1+t (where t=tan(theta) and c = cot(theta))

let theta = say 45 degrees

then

1/0 + 1/0 = 2

Undefined = 2 is not true

the given equation is not an identity

try another angle just in case

let theta = 60 degrees

tan60/(1-cot60) + cot60/(1-tan60) = 1+tan60

sqr3/1/2 +(1/1/2)/(1-sqr3) = 1+sqr3

2sqr3 + 2- 2sqr3 = 1 + sqr3

2 doesn't = 1 +sqr3

it's not an identity

one counterexample proves it.

here's 2 counterexamples. double proof

bigger problem is: how to make it an identity?

change a sign somewhere maybe? 1st possibility

either the problem was miscopied or there was a defect in the original problem itself

if this is a problem from Pearson, that would explain it. They have computer bugs, an infestation