Using standard angle identities, find the value of the product tan 315° × cot(−405°).

Introduction / Context:This problem checks your understanding of trigonometric periodicity and how tangent and cotangent behave for angles beyond 360° and below 0°. You must simplify tan 315° and cot(−405°) using standard reference angles and periodic properties, and then multiply the results. Handling signs correctly is essential when angles lie in different quadrants or are negative.

Given Data / Assumptions:

• The expression to evaluate is tan 315° × cot(−405°).
• Angles are measured in degrees.
• We use exact trigonometric values for the special angle 45°.
• The tangent and cotangent functions have a period of 180°.

Concept / Approach:First, we use the periodicity of tangent and cotangent: tan(θ + 180°) = tan θ and cot(θ + 180°) = cot θ. Next, we reduce 315° and −405° to equivalent angles within a familiar range, such as between 0° and 360° or between −180° and 180°. Both angles will be related to ±45°, for which tan and cot values are well known. Once each value is simplified, we multiply them to obtain the final answer.

Step-by-Step Solution:Start with tan 315°. Since 315° = 360° − 45°, this angle is in the fourth quadrant.Tangent in the fourth quadrant is negative, and the reference angle is 45°, so tan 315° = −tan 45° = −1.Now consider cot(−405°). Use the period of 180° to bring the angle into a standard range.Add 360°: −405° + 360° = −45°. We can also add 180° repeatedly, but cot(−405°) = cot(−45°) is already sufficient.Since cot(−θ) = −cot θ and cot 45° = 1, we have cot(−45°) = −1.Now compute the product: tan 315° × cot(−405°) = (−1) × (−1) = 1.

Verification / Alternative check:An alternative is to convert cotangent into the reciprocal of tangent. Recognise that cot(−405°) = 1 / tan(−405°). First reduce −405° by 180° repeatedly: −405° + 360° = −45°, and tan(−45°) = −1. Therefore cot(−405°) = 1 / (−1) = −1, consistent with the earlier method. With tan 315° = −1 and cot(−405°) = −1, their product is still 1, which confirms the correctness of the result.

Why Other Options Are Wrong:

• −1 would be obtained if only one of the two trigonometric values were negative, but here both are −1.
• 0 is not possible because neither tangent nor cotangent at these angles is zero or undefined.
• 2 and −2 do not arise from multiplying −1 and −1; they would require absolute values greater than 1.

Common Pitfalls:

• Incorrectly reducing −405° to 45° instead of −45°, leading to sign mistakes.
• Confusing the periodicity of 360° with the actual period of tangent and cotangent, which is 180°.
• Forgetting that both tan 315° and cot(−405°) are negative, causing an incorrect negative product.

Final Answer:1