Given: SecA tanA = x

To find: The value of sinA

Solution:

Let's begin by using the trigonometric identity:
Sec^2A = 1 + tan^2A

Since SecA tanA = x, we can rewrite it as:
SecA tanA = SecA * SecA - 1

Now, let's substitute the identity into the equation:
SecA * (1 + tan^2A) = SecA * SecA - 1

Expanding the equation:
SecA + SecA tan^2A = Sec^2A - 1

Rearranging the terms:
SecA tan^2A - SecA + Sec^2A - 1 = 0

This is a quadratic equation in terms of tanA. Let's solve it using the quadratic formula:

tanA = [-(-SecA) ± √((-SecA)^2 - 4(1)(Sec^2A - 1))] / (2(1))

Simplifying further:
tanA = [SecA ± √(Sec^2A - 4(1)(Sec^2A - 1))] / 2

tanA = [SecA ± √(Sec^2A - 4(Sec^2A - 1))] / 2

tanA = [SecA ± √(Sec^2A - 4Sec^2A + 4)] / 2

tanA = [SecA ± √(-3Sec^2A + 4)] / 2

Now, we know that tanA = sinA / cosA. Let's substitute this back into the equation:

sinA / cosA = [SecA ± √(-3Sec^2A + 4)] / 2

Multiplying both sides by cosA:
sinA = [SecA ± √(-3Sec^2A + 4)] * cosA / 2

Explanation:

The value of sinA cannot be determined directly from the given equation SecA tanA = x. However, by using trigonometric identities and solving a quadratic equation, we were able to derive an expression for sinA in terms of SecA and cosA. The solution involves the use of the quadratic formula and substitution of the identity tanA = sinA / cosA.

Summary:

The value of sinA can be expressed as:

sinA = [SecA ± √(-3Sec^2A + 4)] * cosA / 2