Final answer:
To find the cosine from the tangent, one must use trigonometric identities and the Pythagorean theorem. The cosine can be calculated even with the tangent being negative by considering the angle's quadrant.
Step-by-step explanation:
The student seems to be asking about the relationship between the tangent and the cosine functions of angles in trigonometry. To find the cosine of an angle when the tangent is known, one can use the trigonometric identity that relates these functions: tan(\(\theta\)) = sin(\(\theta\))/cos(\(\theta\)). If tan(\(\theta\)) is known and is equal to -\(\sqrt{15}\), the Pythagorean identity can be used: sin2(\(\theta\)) + cos2(\(\theta\)) = 1. Using these identities, we can express sin(\(\theta\)) in terms of tan(\(\theta\)) and solve for cos(\(\theta\)), taking into account that cos(\(\theta\)) can be positive or negative depending on the quadrant in which the angle lies.