Trigonometric Ratios - Introduction to Trigonometry

To understand the relationship between M and N, we need to first define the trigonometric ratios involved.

Trigonometric Ratios:
Trigonometric ratios are ratios of the sides of a right-angled triangle. The three primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan). In this case, we are given the values of secA and tanA.

The secant (sec) of an angle A is the ratio of the hypotenuse to the adjacent side, which can be expressed as:
secA = hypotenuse / adjacent

The tangent (tan) of an angle A is the ratio of the opposite side to the adjacent side, which can be expressed as:
tanA = opposite / adjacent

Calculation:
We are given the values M = secA tanA and N = secA - tanA. Let's substitute the values of secA and tanA to find the relationship between M and N.

Using the trigonometric identities, we can express secA and tanA in terms of the sides of a right-angled triangle.

We know that secA = 1/cosA and tanA = sinA/cosA.

Let's assume a right-angled triangle with angle A. The adjacent side is represented by 'x', the opposite side by 'y', and the hypotenuse by 'z'.

Using the Pythagorean theorem, we have the equation:
x^2 + y^2 = z^2

From the definition of secA and tanA, we can write:
secA = z/x
tanA = y/x

Substituting these values in M and N:

M = secA tanA = (z/x) * (y/x) = (zy) / (x^2)
N = secA - tanA = (z/x) - (y/x) = (z - y) / x

Relationship between M and N:
We can now substitute the values of secA and tanA in terms of the sides of the triangle in M and N:

M = (zy) / (x^2)
N = (z - y) / x

Simplifying these expressions, we get:

M^2 - N^2 = [(zy)^2 / (x^4)] - [(z - y)^2 / x^2]

Further simplifying, we have:

M^2 - N^2 = [(z^2 y^2) - (z^2 - 2zy + y^2)] / (x^2)

Simplifying the numerator:
M^2 - N^2 = [2zy] / (x^2)

Thus, we can conclude that M^2 - N^2 = 2zy / x^2, which establishes the relationship between M and N.

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