Trigonometric Identity for Acute Angles

The question asks for a trigonometric identity that holds true for all acute angles A. An acute angle is an angle greater than 0 degrees and less than 90 degrees ($0^\circ < A < 90^\circ$). We need to evaluate the given options based on fundamental trigonometric identities.

Understanding Co-function Identities

The options involve complementary angles, specifically angles of the form $90^\circ - A$. The key co-function identities are:

• $\sin(90^\circ - A) = \cos A$
• $\cos(90^\circ - A) = \sin A$
• $\tan(90^\circ - A) = \cot A$
• $\cot(90^\circ - A) = \tan A$

These identities are true for all angles.

Evaluating the Options

Let's examine each option:

1. $\sin A = \sin(90^\circ - A)$

   Using the co-function identity, $\sin(90^\circ - A) = \cos A$. So, this option becomes $\sin A = \cos A$. This is only true for $A = 45^\circ$, not for *all* acute angles.

2. $\sin A = \cos(90^\circ - A)$

   According to the co-function identities, $\cos(90^\circ - A)$ is indeed equal to $\sin A$. This identity is true for all angles, including all acute angles A.

3. $\cos A = \tan(90^\circ - A)$

   Using the co-function identity, $\tan(90^\circ - A) = \cot A$. So, this option becomes $\cos A = \cot A$. This is only true for $A = 45^\circ$, not for *all* acute angles.

4. $\cot A = \sin(90^\circ - A)$

   Using the co-function identity, $\sin(90^\circ - A) = \cos A$. So, this option becomes $\cot A = \cos A$. This is generally not true for acute angles. For instance, if $A = 30^\circ$, $\cot 30^\circ = \sqrt{3}$ and $\cos 30^\circ = \frac{\sqrt{3}}{2}$. These are not equal.

Conclusion

Based on the analysis of the co-function identities, the only statement that is true for all acute angles A is $\sin A = \cos(90^\circ - A)$.