Prove the identity cot x + tan x = sec x cosec x

Answer 1

We begin by using the definitions of tangent, cotangent, secant, and cosecant:

cot x = cos x / sin x

tan x = sin x / cos x

sec x = 1 / cos x

cosec x = 1 / sin x

Substituting these definitions into the left-hand side of the identity, we get:

cot x + tan x = cos x / sin x + sin x / cos x

To add these fractions, we need a common denominator:

cot x + tan x = (cos x * cos x) / (cos x * sin x) + (sin x * sin x) / (cos x * sin x)

Now we can combine the numerators:

cot x + tan x = (cos^2 x + sin^2 x) / (cos x * sin x)

Using the Pythagorean identity, we know that cos^2 x + sin^2 x = 1, so we can simplify:

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

Using the definitions of secant and cosecant, we see that the right-hand side of the identity is:

sec x cosec x = (1 / cos x) * (1 / sin x) = 1 / (cos x * sin x)

Therefore, we have shown that:

cot x + tan x = sec x cosec x

as required.