Inverse Trigonometric Functions

Put x = cos\,y y \,\le\,\pi \left|x\right| \le \,1 \,\,\,\,\,\dots(i)\) 
and the RHS of given equation becomes 

Since, lies between and
2y lies between and . 
i.e., y lies between - and . 

On combining Eqs. (i) and (ii), we get 

Because they are the inverse of trigonometric functions, inverse trigonometric functions are often referred to as anti-trigonometric functions. The ''Arc Functions'' are another name for them. The trigonometric functions are reversed by inverse trigonometric functions. When any two sides of a right angled triangle are known, these functions can be used to calculate the angles. The prefix 'arc' is used to identify the inverse trigonometric functions.

Trigonometric functions have an inverse known as an inverse trigonometric function. Sine, cosine, tangent, cotangent, secant, and cosecant are the fundamental trigonometric functions. Arcus functions, anti trigonometric functions, and cyclometer functions are further names for these operations. Numerous disciplines, including physics, engineering, geometry, navigation, aviation, marine biology, etc., use inverse trigonometry. Any trigonometric function may be used to calculate the angles of a triangle using inverse trigonometry. 

Inverse Functions: Domain and Range 

sin(sin⁻¹ x) = x, if -1 ≤ x ≤ 1

cos(cos⁻¹x) = x, if -1 ≤ x ≤ 1

tan(tan⁻¹x) = x, if -∞ ≤ x ≤∞

cot(cot⁻¹x) = x, if -∞≤ x ≤∞

sec(sec⁻¹x) = x, if -∞ ≤ x ≤ -1 or 1 ≤ x ≤ ∞

cosec(cosec⁻¹x) = x, if 1 ≤ x ≤ ∞ or -∞ ≤ x ≤ -1

Inverse Trigonometric Functions: Domain and Range 

In case of Inverse Trigonometric Functions, the formulas are:

sin⁻¹(sin y) = y, if -π/2 ≤ y ≤ π/2

cos⁻¹(cos y) = y, if 0 ≤ y ≤ π

tan⁻¹(tan y) = y, if -π/2 < y < π/2

cot⁻¹(cot y) = y if 0 < y < π

sec⁻¹(sec y) = y, if 0 ≤ y ≤ π, y ≠ π/2

cosec⁻¹(cosec y) = y, if -π/2 ≤ y ≤ π/2, y ≠ 0

If , then we need to find .

We recognize that the expression inside the can be written as , let so

Now we find the derivative :

Now we evaluate :

Therefore, the correct option is (A) .

We need to evaluate the expression:

Let's evaluate the two main parts of the expression separately.

Part 1:

First, evaluate :

Since has a period of :

Now substitute this back into the first part:

The principal value range for is . The angle in this range such that is .

So, Part 1 evaluates to .

Part 2:

First, evaluate the innermost term :

The principal value range for is . The angle in this range such that is .

Now substitute this back into the second part:

Evaluate :

Substitute this back:

The angle in the range such that is .

So, Part 2 evaluates to .

Combining the Parts:

The original expression shows the two parts written next to each other: . In standard mathematical notation, juxtaposition implies multiplication.

However, is not among the options (A) 0, (B) , (C) , (D) .

This suggests there might be a typo in the question, possibly an omitted operator. If we assume the intended operation between the two parts was addition instead of multiplication:

This result, , matches option (C).

Assuming addition was intended due to the options provided:

The final answer is