If cosec a cot a = m, prove that (m^2 - 1)/(m^2 + 1) = cos^2(a) - sin^2(a).

Proof:

To prove that (m^2 - 1)/(m^2 + 1) = cos^2(a) - sin^2(a), let's start by expressing cosec a and cot a in terms of sin a and cos a.

Expressing cosec a and cot a in terms of sin a and cos a:

Using the definitions of cosec a and cot a, we have:

cosec a = 1/sin a
cot a = cos a/sin a

Substituting the expressions of cosec a and cot a:

Now, substituting the values of cosec a and cot a in the given equation m = cosec a cot a, we get:

m = (1/sin a) * (cos a/sin a)

Simplifying the expression:

m = cos a/sin^2 a

Squaring both sides of the equation:

Now, let's square both sides of the equation m = cos a/sin^2 a to eliminate the square in the denominator:

m^2 = (cos^2 a)/(sin^4 a)

Using the identity sin^2 a + cos^2 a = 1:

We know that sin^2 a + cos^2 a = 1. Rearranging the equation, we get:

sin^2 a = 1 - cos^2 a

Substituting this expression in the equation m^2 = (cos^2 a)/(sin^4 a), we get:

m^2 = cos^2 a/(1 - cos^2 a)^2

Expanding the denominator:

Expanding the denominator (1 - cos^2 a)^2, we get:

m^2 = cos^2 a/(1 - 2cos^2 a + cos^4 a)

Simplifying the expression:

To simplify the expression further, we can use the identity cos^2 a = 1 - sin^2 a. Substituting this in the equation, we get:

m^2 = (1 - sin^2 a)/(1 - 2(1 - sin^2 a) + (1 - sin^2 a)^2)

Simplifying the expression:

m^2 = (1 - sin^2 a)/(1 - 2 + 2sin^2 a - sin^4 a)

m^2 = (1 - sin^2 a)/(3 - 2sin^2 a + sin^4 a)

Using the identity sin^2 a + cos^2 a = 1:

We know that sin^2 a + cos^2 a = 1. Rearranging the equation, we get:

cos^2 a = 1 - sin^2 a

Substituting this expression in the equation m^2 = (1 - sin^2 a)/(3 - 2sin^2 a + sin^4 a), we get:

m^2 = cos^2 a/(3 - 2sin^2 a + sin^4 a)

Using the identity cos^2