Express cot 85° + cos 75° in terms of trigonometric ratios of angles between 0° and 45°.

To express $\cot 85^\circ + \cos 75^\circ$ in terms of trigonometric ratios of angles between $0^\circ$ and $45^\circ$, we will utilize trigonometric identities.

Step 1: Break Down the Problem

We need to express $\cot 85^\circ$ and $\cos 75^\circ$ in terms of angles less than $45^\circ$.

Step 2: Relevant Concepts

1. Cotangent Identity: $\cot(90^\circ - x) = \tan(x)$
2. Cosine Identity: $\cos(90^\circ - x) = \sin(x)$

Step 3: Analyze and Detail

1. Calculate $\cot 85^\circ$: $\cot 85^\circ = \tan(90^\circ - 85^\circ) = \tan 5^\circ$

2. Calculate $\cos 75^\circ$: $\cos 75^\circ = \sin(90^\circ - 75^\circ) = \sin 15^\circ$

Therefore, we can rewrite the expression: $\cot 85^\circ + \cos 75^\circ = \tan 5^\circ + \sin 15^\circ$

Step 4: Verify and Summarize

The expressions $\tan 5^\circ$ and $\sin 15^\circ$ are both in the range of angles between $0^\circ$ and $45^\circ$.

Final Answer

Thus, the final expression is: $\tan 5^\circ + \sin 15^\circ$