A) cos θ = -(√24)/7 and tan θ = 5/√24 and
B) sec θ = 7/(√24) and tan θ = -5/(√24)
Sine is the ratio of the opposite side of an angle to the hypotenuse. Since sin θ = -5/7, the opposite side is -5 and the hypotenuse is 7.
For A, if the cosine is -√24/7, and the sine was -5/7, then the opposite/adjacent, tangent, would be -5/-(√24). However, two negatives divided make a positive, so the tangent would be 5/√24.
For B: the secant is 1/cos. This means it is the reciprocal, or flip, of the cosine. Instead of A/H, sec = H/A. This means the hypotenuse is 7 and the adjacent side is √24. Tangent = opposite/adjacent; this gives us -5/√24.
For C: the secant, H/A, is -7/5. This means the tangent, O/A, would be -5/5 = -1. This is incorrect.
For D: the cosine, A/H, is √24/7. This means the tangent, O/A, would be -5/√24; this is incorrect.