In right angled triangle ΔXYZ, the right angle is at vertex Y. If tan X = 24/7, use the complementary angle relationship to find the exact value of cot Z.

Introduction / Context:
This trigonometry question uses properties of a right angled triangle and the relationship between acute angles in such a triangle. In a right triangle, the two acute angles are complementary, which links their trigonometric ratios. Here, you are given tan X and asked to find cot Z in triangle ΔXYZ with a right angle at Y. This is a standard type of question that tests understanding of basic trigonometric identities and complementary angle concepts.

Given Data / Assumptions:

• Triangle ΔXYZ is right angled at Y.
• Angles X and Z are acute angles.
• tan X = 24/7.
• We must find cot Z.
• All sides are positive real lengths.

Concept / Approach:
In a right angled triangle, the sum of the acute angles X and Z is 90°. Therefore Z = 90° − X. Trigonometric ratios of complementary angles are related: tan X = cot (90° − X) and cot X = tan (90° − X). Since Z and X are complementary, tan X and cot Z are equal. This allows direct substitution instead of recomputing side ratios from scratch.

Step-by-Step Solution:
Step 1: Since ΔXYZ is right angled at Y, we have X + Z = 90°.Step 2: Therefore Z = 90° − X, so Z is the complementary angle of X.Step 3: Recall the identity tan X = cot (90° − X) for acute angles.Step 4: Substitute Z = 90° − X into the identity to obtain tan X = cot Z.Step 5: The problem states tan X = 24/7.Step 6: Hence cot Z = 24/7.

Verification / Alternative check:
We can visualise the sides of ΔXYZ. For angle X, tan X = opposite side over adjacent side. If we take opposite side to X as 24k and adjacent side to X as 7k for some positive k, then by the Pythagorean theorem the hypotenuse is 25k. For angle Z at the other acute vertex, the roles of the legs are reversed. The cotangent of Z is adjacent over opposite relative to angle Z, which works out as 24k/7k = 24/7, again confirming the result.

Why Other Options Are Wrong:
The value 7/24 is equal to tan Z, not cot Z, and results from inverting tan X instead of using the complementary angle relationship. The fractions 25/7 and 24/25 involve the hypotenuse and are not correct for cot Z. The option 25/24 is another distractor based on mixing leg and hypotenuse lengths. Only 24/7 matches the correct identity tan X = cot Z for complementary angles.

Common Pitfalls:
Many learners mistakenly compute cot Z as 1/tan X, obtaining 7/24, which corresponds to cot X rather than cot Z. Another common error is to forget that in a right triangle, the two acute angles are complementary, and therefore their trigonometric ratios are related through specific identities. Remembering tan X = cot (90° − X) and carefully identifying which angle is complementary helps avoid these mistakes.

Final Answer:
The required value of cot Z in triangle ΔXYZ is 24/7.