Proof: √sec^2(A) * √cosec^2(A) = tan(A) * cot(A)

To prove the given equation, we will start by simplifying both sides of the equation separately and then show that they are equal.

Simplifying the left-hand side (LHS):

Let's start by simplifying the left-hand side of the equation.

LHS = √sec^2(A) * √cosec^2(A)

Recall that the square root (√) is the inverse operation of squaring. Therefore, the square root of the square of any number is the absolute value of that number.

So, we can rewrite the left-hand side as:

LHS = |sec(A)| * |cosec(A)|

Now, we need to express sec(A) and cosec(A) in terms of other trigonometric functions.

Recall the definitions of secant (sec) and cosecant (cosec):

sec(A) = 1/cos(A)

cosec(A) = 1/sin(A)

Substituting these values into our expression, we get:

LHS = |1/cos(A)| * |1/sin(A)|

Taking the absolute value of a fraction is the same as taking the absolute value of the numerator and denominator separately. So, we can rewrite the expression as:

LHS = |1|/|cos(A)| * |1|/|sin(A)|

LHS = 1/(|cos(A)| * |sin(A)|)

Now, we know that |cos(A)| * |sin(A)| is equal to sin(A) * cos(A) since both cos(A) and sin(A) are positive in the first and second quadrants.

Therefore, LHS simplifies to:

LHS = 1/(sin(A) * cos(A))

Simplifying the right-hand side (RHS):

Now, let's simplify the right-hand side of the equation.

RHS = tan(A) * cot(A)

Recall the definitions of tangent (tan) and cotangent (cot):

tan(A) = sin(A)/cos(A)

cot(A) = 1/tan(A) = cos(A)/sin(A)

Substituting these values into our expression, we get:

RHS = (sin(A)/cos(A)) * (cos(A)/sin(A))

Notice that sin(A) cancels out with sin(A) and cos(A) cancels out with cos(A), leaving us with:

RHS = 1

Comparing LHS and RHS:

Now that we have simplified both sides of the equation, we can compare them:

LHS = 1/(sin(A) * cos(A))

RHS = 1

We can see that LHS and RHS are equal, thus proving the given equation:

√sec^2(A) * √cosec^2(A) = tan(A) * cot(A)