If secθ + tanθ = x, then find sinθ.

Question

Accepted Answer

Given: secθ + tanθ = x

We know the identity:
(secθ + tanθ)(secθ - tanθ) = 1

Therefore,
secθ - tanθ = 1/x

Adding the two equations:
(secθ + tanθ) + (secθ - tanθ) = x + 1/x
2 secθ = x + 1/x
secθ = (x + 1/x)/2

Subtracting the two equations:
(secθ + tanθ) - (secθ - tanθ) = x - 1/x
2 tanθ = x - 1/x

tanθ = (x - 1/x)/2

Recall:
secθ = 1/cosθ
sinθ = tanθ * cosθ

So,
sinθ = tanθ / secθ = ( (x - 1/x)/2 ) / ( (x + 1/x)/2 ) = (x - 1/x) / (x + 1/x)

Multiply numerator and denominator by x to clear fractions:
= (x^2 - 1) / (x^2 + 1)

Hence,
sinθ = (x^2 - 1) / (x^2 + 1)

This matches option 3.

Answer

Answer

Answer