Prove the identity cot x + tan x = sec x cosec x
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.
To prove the identity cot x + tan x = sec x cosec x, we need to manipulate the left side of the equation until it is equivalent to the right side of the equation.
We'll start with the left side of the equation, cot x + tan x.
First, let's express cot x and tan x in terms of sin x and cos x:
cot x = cos x / sin x
tan x = sin x / cos x
Now, we can substitute these expressions into the left side of the equation:
cot x + tan x = cos x / sin x + sin x / cos x
To add these fractions, we need to obtain a common denominator. The common denominator is sin x * cos x, so we'll multiply the first fraction by cos x / cos x and the second fraction by sin x / sin x:
cot x + tan x = cos x * cos x / (sin x * cos x) + sin x * sin x / (sin x * cos x)
Simplifying the fractions:
cot x + tan x = cos^2 x / (sin x * cos x) + sin^2 x / (sin x * cos x)
Now, let's combine the fractions with the same denominators:
cot x + tan x = (cos^2 x + sin^2 x) / (sin x * cos x)
The identity cos^2 x + sin^2 x = 1 holds for all values of x, so we can simplify further:
cot x + tan x = 1 / (sin x * cos x)
Finally, we can use the reciprocal identities to express 1 / (sin x * cos x) as sec x * cosec x:
cot x + tan x = sec x * cosec x
Therefore, we have proven that cot x + tan x = sec x * cosec x.