The trigonometric ratios cosine and cotangent are reciprocals of each other true or false

That's FALSE.

In the first quadrant the co functions decrease with increasing angles while the non-co functions increase with angle in the first quadrant.

Since reciprocals move in the opposite direction, each reciprocal pair is a co functions and a non co function. So we're FALSE here since we have two co functions, which can't be reciprocals.

In our case, the reciprocal of cosine is secant, and the reciprocal of cotangent is tangent.