Help with trigonometry homework

By definition of tangent,

tan(A - π/4) = sin(A - π/4) / cos(A - π/4)

Expand the numerator and denominator using the angle sum identities for sin and cos:

tan(A - π/4) = (sin(A) cos(π/4) - cos(A) sin(π/4)) / (cos(A) cos(π/4) + sin(A) sin(π/4))

Divide through everything on the right by cos(A) cos(π/4):

tan(A - π/4) = (sin(A) / cos(A) - sin(π/4) / cos(π/4)) / (1 + (sin(A) sin(π/4)) / (cos(A) cos(π/4)))

Simplify the sin/cos terms to tan:

tan(A - π/4) = (tan(A) - tan(π/4)) / (1 + tan(A) tan(π/4))

tan(π/4) = 1, so we're left with

tan(A - π/4) = (tan(A) - 1) / (1 + tan(A))

Replace tan(A) with -√15:

tan(A - π/4) = (-√15 - 1) / (1 - √15)

Then the last option is the correct one.

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step-by-step explanation:

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