Question

(cot(x)-tan(x))/(sin(x)cos(x)) = csc^2(x)-sec^(2)x

Answer 1

i need to prove it

Answer 2

rewrite cot ans tan in terms of sin and cos

\[\frac{\frac{\cos(x)}{\sin(x)} - \frac{\sin(x)}{\cos(x)}}{\sin(x)\cos(x)} = \frac{\frac{\cos^2(x) - \sin^2(x)}{\sin(x)\cos(x)}}{\sin(x)\cos(x)} \]

this can be simplified to

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

now just split the fraction

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

just remove the common factors for the answer.

Answer 3

cot x = cosx/sinx
tan x = sinx/cosx

Take Left side first
(cosx/sinx - sinx/cosx) (1/sinx cosx)

Now take common denominator

(cos^2 x - sin^2 x)/sinx cosx (1/sinx cosx)

Multiply the common denominator with the denominator give in the equation
(cos^ 2 x - sin^2 x) / sin^2 x cos^ 2x

Separate both terms
(cos^2 x/cos^2 x sin^2 x) - (sin^2 x/ sin^2 x cos^2 x)

Cancel out cosine and sines

1/sin^2 x - 1/cos^2 x
csc^2 x - sec^2 x which is equal to the right side