Evaluating a trigonometric quotient with angle identities

Key concept: This expression is best handled by rewriting each trig function using angle-sum and angle-subtraction identities.

Step by step

Key idea

Simplify each trig factor separately using standard identities:

• $\cot\left(\frac{3\pi}{2}-\alpha\right)$
• $\cos\left(\frac{\pi}{2}+\alpha\right)$
• $\tan(\pi-\alpha)$
• $\sin(\pi-\alpha)$

Then combine the results carefully.

Step-by-step simplification

Use the identities:

$\cot\left(\frac{3\pi}{2}-\alpha\right)=\tan\alpha,$

$\cos\left(\frac{\pi}{2}+\alpha\right)=-\sin\alpha,$

$\tan(\pi-\alpha)=-\tan\alpha,$

$\sin(\pi-\alpha)=\sin\alpha.$

Substitute these into the given quotient:

=\frac{(\tan\alpha)(-\sin\alpha)}{(-\tan\alpha)(\sin\alpha)}.$$ Now simplify the signs and cancel common factors: $$\frac{-\tan\alpha\sin\alpha}{-\tan\alpha\sin\alpha}=1.$$ So the whole expression simplifies to $$\boxed{1}.$$ ## Why this works Each angle is a standard shift by $\pi$ or $\frac{\pi}{2}$, so the exact identities are well known. The key is not to expand everything from scratch; instead, recognize the quadrant behavior and sign changes. Since both numerator and denominator contain the same factors after simplification, everything cancels. ## Common checks Before canceling, confirm that no factor is zero. In a formal proof or exam solution, it is enough to show the identity chain and then reduce the quotient. The final result is independent of $\alpha$ wherever the original expression is defined. ### Pitfall alert A common mistake is mixing up sine and cosine signs for angle shifts by $\frac{\pi}{2}$ and $\pi$. Another trap is canceling terms before rewriting them in a comparable form. For example, $\cot(\frac{3\pi}{2}-\alpha)$ does not look cancelable until you convert it using identities. Also, remember that division by zero is not allowed, so the expression is only valid where both the numerator and denominator are defined. Working symbol by symbol without identities usually leads to sign errors. ### Try different conditions If the expression were $$\frac{\tan(\frac{3\pi}{2}-\alpha)\cdot\sin(\frac{\pi}{2}+\alpha)}{\cot(\pi-\alpha)\cdot\cos(\pi-\alpha)},$$ you would use a different set of identities, but the same method would apply: rewrite each term, simplify the signs, and cancel common factors. A small change in the angle or function can flip a sign or turn a sine into a cosine, so the exact identity list matters more than memorizing a single final answer. ### Further reading Angle identities, Cotangent, Reference angle