Question

Prove the identity.cot x (sec^2x-1)= tan x

Answer 1

$cot\text{ }x(sec^2x-1)=tan\text{ }x$

Using one of the Pythagorean Identities, we can replace sec²x with 1 + tan²x. The equation above becomes:

$cot\text{ }x(1+tan^2x-1)=tan\text{ }x$

Applying Algebra, we can add the terms 1 and -1 in the parenthesis. The equation becomes:

$cot\text{ }x(tan^2x)=tan\text{ }x$

Applying the reciprocal of cot x, we can replace cot x with 1/tan x.

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

Divide tan²x by tan x and the quotient is:

$tan\text{ }x=tan\text{ }x$

As we can see above, the left side of the equation has been proven to be equal to its right side. Hence, the trigonometric equation is an identity.