Proof/Identity Check in Degrees

The correct identity for cos(105°) is derived from the cosine of a sum formula, which states that cos(A + B) = cosA * cosB - sinA * sinB. In this case, 105° can be expressed as 60° + 45°, thus making option A the correct choice. The other options either involve incorrect angle sums or differences that do not evaluate to 105°.

The correct identity for sin(15°) can be derived using the sine subtraction formula. The formula states that sin(a - b) = sin(a)cos(b) - cos(a)sin(b). For sin(15°), we can use 45° and 30° (15° = 45° - 30°). According to the formula, sin(15°) = sin(45°)cos(30°) - cos(45°)sin(30°). Thus, option D is the correct identity.

The identity used in option B is the cosine addition formula, which accurately breaks down the angle into its components and calculates the cosine based on known values. Specifically, cos(165°) can be evaluated as cos(120° + 45°) using the addition formula, confirming that option B is correct.

To find tan(105°), we can use the angle addition identity. The correct choice is D, which represents the sum of angles tan(60° + 45°). This identity is valid because tan(θ + φ) = (tanθ + tanφ) / (1 - tanθtanφ), and using 60° and 45° leads us to the correct calculation of tan(105°). The other options do not correctly apply the tangent identities for the specified angles.

The identity used in option A, sin(180° - 15°) = sin(15°), is a standard sine identity that states that the sine of an angle and the sine of its supplementary angle are equal. In this case, 165° can be expressed as 180° - 15°, confirming that option A is correct.

The correct identity for cos(15°) can be derived using the cosine subtraction formula: cos(a - b) = cos(a)cos(b) + sin(a)sin(b). In this case, when using cos(30° - 15°), we have a = 30° and b = 15°, making option C the correct representation. The other options represent different angles and thus do not equate to cos(15°).

The correct identity for tan(165°) is found using the identity for tangent of an angle in the second quadrant. Since 165° is in the second quadrant, we can express it as 180° - 15°, which gives us tan(165°) = -tan(15°). Therefore, option A is correct as it follows the tangent subtraction formula for angles.

The correct identity for sin(11π/12) is derived from the sine addition formula. In option D, we can see that sin(3π/4 + π/6) correctly uses the sine addition property. Calculation shows that sin(3π/4) and cos(π/6), along with the other terms, yield the correct value for sin(11π/12), confirming that option D is valid.

The identity used in option A leverages the cosine subtraction formula, which states that cos(a - b) = cos(a)cos(b) + sin(a)sin(b). Since 11π/12 is equivalent to π - π/12, we can rewrite it as cos(π - π/12), which leads to -cos(π/12) due to the properties of cosine in different quadrants.

The identity for tan(11\π/12) can be evaluated using the property of tangent that relates angles. Specifically, tan(11\π/12) can be expressed as tan(\π - \π/12), which simplifies to -tan(\π/12) based on the tangent's odd function property. Therefore, option A is the correct identity.

The correct identity for sin(5π/12) can be derived using the sine addition formula. The angle 5π/12 can be expressed as π/4 + π/6. According to the sine addition formula, sin(a + b) = sin(a)cos(b) + cos(a)sin(b). Here, using a = π/4 and b = π/6, the identity D correctly represents this relationship, confirming that sin(5π/12) = sin(π/4)cos(π/6) + cos(π/4)sin(π/6).

The correct identity for cos(500π/12) is A, which represents the cosine sum formula. By using the cosine addition formula, we can break down the angle into two known angles (π/4 and π/6) to compute the cosine of the sum.

The correct identity for tan(5π/12) can be derived using the angle addition formula for tangent, which states that tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b)). In this case, 5π/12 can be expressed as π/4 + π/6, making option D the correct choice as it properly applies the formula for the addition of angles.

The cosine function is an even function, which means that cos(-x) = cos(x) for any angle x. Therefore, cos(-15°) is equal to cos(15°), making option D the correct answer. The other options represent different angle identities that do not apply to this specific case.

The correct identity for sin(75°) is found using the angle addition formula, which states that sin(A + B) = sin(A)cos(B) + cos(A)sin(B). In this case, 75° can be expressed as 45° + 30°, making option C the correct choice. The other options do not correctly represent sin(75°) using valid angle identities.

The correct identity for cos(15°) can be derived using the cosine subtraction formula: cos(A - B) = cosA*cosB + sinA*sinB. In option D, substituting A = 45° and B = 30°, we have cos(45° - 30°) = cos(45°) * cos(30°) + sin(45°) * sin(30°), which simplifies to the correct expression for cos(15°).

The formula for the tangent of a difference of angles states that tan(A - B) = (tanA - tanB) / (1 + tanA * tanB). In this case, tan(15°) can be expressed as tan(45° - 30°). Substituting A as 45° and B as 30°, we find that option A correctly applies this formula, confirming it is the correct identity.

The correct identity for cos(75°) can be derived from the angle addition formula for cosine, which states that cos(A + B) = cos(A)cos(B) - sin(A)sin(B). In option D, substituting A = 45° and B = 30° gives cos(75°) = cos(45°)cos(30°) - sin(45°)sin(30°), which is correct. The other options either misuse the angle addition formulas or misrepresent the identities.

The correct identity for sin(105°) is given by the angle addition formula, sin(a + b) = sin(a)cos(b) + cos(a)sin(b). In this case, 60° + 45° correctly adds up to 105°, making option B the right choice. Other options use different angle identities that do not yield sin(105°).

The correct identity for tan(75°) uses the angle addition formula for tangent: tan(A + B) = (tanA + tanB) / (1 - tanAtanB). In this case, 75° can be expressed as 45° + 30°, making option D the right choice since it accurately applies this formula.