Evaluate any inverse trigonometric function in its principal branch, add two arctangents with the correct branch correction, and check the whole Class 12 Chapter 2 formula sheet in one place.
Every trigonometric function repeats, so none of them has an inverse until its domain is cut down to one stretch where it is one-to-one. That chosen stretch becomes the range of the inverse, and it is the reason sin⁻¹(1/2) is π/6 and not also 5π/6.
| Function | Domain | Range (principal branch) |
|---|---|---|
| sin⁻¹ x | [−1, 1] | [−π/2, π/2] |
| cos⁻¹ x | [−1, 1] | [0, π] |
| tan⁻¹ x | ℝ (all real numbers) | (−π/2, π/2) |
| cot⁻¹ x | ℝ (all real numbers) | (0, π) |
| sec⁻¹ x | (−∞, −1] ∪ [1, ∞) | [0, π] − {π/2} |
| cosec⁻¹ x | (−∞, −1] ∪ [1, ∞) | [−π/2, π/2] − {0} |
Note the pattern: the three “co” functions (cos, cot, cosec) and their partners pair up, and each pair's ranges are what make the complementary identities below come out to π/2.
sin⁻¹x + cos⁻¹x = π/2 · tan⁻¹x + cot⁻¹x = π/2 · sec⁻¹x + cosec⁻¹x = π/2
sin⁻¹(1/x) = cosec⁻¹x ·
cos⁻¹(1/x) = sec⁻¹x ·
tan⁻¹(1/x) = cot⁻¹x for x > 0
sin⁻¹(−x) = −sin⁻¹x ·
tan⁻¹(−x) = −tan⁻¹x ·
cos⁻¹(−x) = π − cos⁻¹x ·
cot⁻¹(−x) = π − cot⁻¹x
tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) only if xy < 1
= π + tan⁻¹((x+y)/(1−xy)) if xy > 1 and x, y > 0
= −π + tan⁻¹((x+y)/(1−xy)) if xy > 1 and x, y < 0
cos⁻¹x + cos⁻¹y = cos⁻¹(xy − √(1−x²)·√(1−y²)) if x + y ≥ 0
2 tan⁻¹x = tan⁻¹(2x/(1−x²)) if |x| < 1
2 tan⁻¹x = sin⁻¹(2x/(1+x²)) if |x| ≤ 1
2 tan⁻¹x = cos⁻¹((1−x²)/(1+x²)) if x ≥ 0
| y | dy/dx | Valid for |
|---|---|---|
| sin⁻¹ x | 1 / √(1 − x²) | |x| < 1 |
| cos⁻¹ x | −1 / √(1 − x²) | |x| < 1 |
| tan⁻¹ x | 1 / (1 + x²) | all x |
| cot⁻¹ x | −1 / (1 + x²) | all x |
| sec⁻¹ x | 1 / (|x| √(x² − 1)) | |x| > 1 |
| cosec⁻¹ x | −1 / (|x| √(x² − 1)) | |x| > 1 |
Each “co” derivative is exactly the negative of its partner, which follows immediately from differentiating the complementary identities — if sin⁻¹x + cos⁻¹x is the constant π/2, the two derivatives must cancel.
cos⁻¹x falls from π to 0 across [−1, 1]; sin⁻¹x rises from −π/2 to π/2 across the same interval; tan⁻¹x is defined for every real x and flattens towards the horizontal asymptotes at ±π/2 without ever reaching them.
The 17s cancelling is the whole point of the question — it is built so the messy fractions collapse to a clean 1/2.
This is CBSE Class 12 Maths Chapter 2 in full. The chapter is short and it is scoring, because almost every question is one of three types: state a principal value, simplify a composition such as tan⁻¹x + tan⁻¹y, or differentiate. The single most common mistake is applying the arctangent sum formula without first checking whether xy < 1 — that check is worth a mark on its own and the calculator above shows you which branch it used and why.
The domain of cos inverse x is the closed interval from minus one to one. Outside that, there is no angle whose cosine equals the input, so the function is undefined. Its range is 0 to pi.
The domain of sin inverse x is the closed interval from minus one to one, the same as cos inverse. The range differs: sin inverse returns values from minus pi by two to pi by two.
Two tan inverse x equals tan inverse of 2x over 1 minus x squared when the absolute value of x is less than one. It also equals sin inverse of 2x over 1 plus x squared when the absolute value of x is at most one, and cos inverse of 1 minus x squared over 1 plus x squared when x is at least zero.
It equals tan inverse of one half, which is about 0.4636 radians or 26.57 degrees. Because the product of one quarter and two ninths is one eighteenth, which is less than one, the plain sum formula applies without any pi correction.
The derivative of cos inverse x is minus one over the square root of one minus x squared, valid where the absolute value of x is less than one. It is the exact negative of the derivative of sin inverse x.
The derivative of cot inverse x is minus one over one plus x squared, valid for every real x. It is the negative of the derivative of tan inverse x.
Trigonometric functions repeat, so infinitely many angles share the same sine or cosine. Restricting the output to one agreed interval, called the principal branch, is what makes the inverse a genuine function with one answer per input.
Inverse trigonometric functions are Chapter 2 of the CBSE Class 12 Maths syllabus, following Relations and Functions. The chapter covers principal values, the domain and range table, the standard identities and differentiation.