Trigonometric Identities

The tangent function is defined as tan x = sin x / cos x, which is the ratio of sine to cosine for the same angle. Options A and B are incorrect reciprocals, while D would be cotangent (cot x = cos x / sin x).

The secant function is defined as sec x = 1/cos x, the reciprocal of cosine. Option D (1/sin x) would be cosecant (csc x), and options A and B are incorrect expressions. This is a fundamental reciprocal identity in trigonometry.

tan(45°) = sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1. At 45 degrees, sine and cosine have equal values, so their ratio is 1. Infinity would mean cos(45°) = 0, which is false.

The expression simplifies using trigonometric identities: (1-cos x)/(1+cos x) = tan^2(x/2). Thus, y = tan^-1(tan(x/2)) = x/2. The first derivative dy/dx is 1/2, and the second derivative d^2y/dx^2 is 0. The stored answer correctly identifies the second derivative.

To solve for p, first evaluate tan(45°) = 1. Then the equation becomes 3(1) + p = 8, which simplifies to 3 + p = 8. Subtracting 3 from both sides gives p = 5. This is a basic trigonometry substitution problem combined with linear equation solving.

The derivative of cos(x) is -sin(x). This follows from the rules of trigonometric differentiation. Conversely, the derivative of sin(x) is positive cos(x). Distractors like -cos(x) or -tan(x) do not follow the standard differentiation rules for this function.

The derivative of tan x is sec^2 x. This follows from writing tan x = sin x/cos x and applying the quotient rule. The notation 'sec2 x' in option A is meant to represent sec²(x). Options B and C use csc (cosecant), which is unrelated to tan derivatives. Option D has the wrong sign.

Cot(90°) equals 0, which is a defined value. The other three trigonometric expressions are undefined because they involve division by zero: tan(90°) = sin(90°)/cos(90°) = 1/0, cosec(0°) = 1/sin(0°) = 1/0, and sec(90°) = 1/cos(90°) = 1/0.

tan 45 = 1, sin 90 = 1, and cos 0 = 1. Adding them together: 1 + 1 + 1 = 3.

sin 45 = 0.707 and tan 45 = 1. The inequality 0.707 > 1 is false, so the correct answer is False.

tan 30 = 1/sqrt(3) and tan 60 = sqrt(3). Their product is 1. sin 90, cos 0, and tan 45 are all equal to 1. Therefore, all marked options are mathematically correct.

Evaluating: tan 60 = 1.732, cos 45 = 0.707, sin 45 = 0.707, tan 30 = 0.577. Sum = 3.723, which is greater than 0, 1, 2, and 3 but less than 4. Therefore options A, B, C, and D are all correct.

If tan(2x) = 4sqrt(2)/7, use tan(2x) = 2tan(x)/(1-tan²x) to find tan(x) = sqrt(2)/4. Then sin(x) = tan(x)/sqrt(1+tan²x). Calculation: sin(x) = (sqrt(2)/4) / sqrt(1 + 2/16) = (sqrt(2)/4) / (3/sqrt(8)) = (sqrt(2)/4) * (2sqrt(2)/3) = 4/12 = 1/3.

The identity tan θ = 1/(cot θ) is true because cot θ is defined as 1/tan θ. Therefore, tan θ = 1/(cot θ) is mathematically correct for all values where both functions are defined. This reciprocal relationship is fundamental in trigonometry.

This mathematical function is used to return arc tangent of the given expression x/y.