Question

$2 \cos^{-1} x = \sin^{-1} ({2x \sqrt {1-x^2)}}$ is valid for all values of $x$ satisfying

• A. $-1 \leq x\leq1 $
• B. $0\leq x\leq1 $
• C. $\frac{1}{\sqrt {2}} \leq x \leq 1$
• D. $0 \leq x \leq\frac{1}{\sqrt {2}} $

Answer 1

The Correct Option is C

Put \(cos^{-1}\, x = y, so that\)x = cos\,y$ 
Then , \(0 \,\le\) y \,\le\,\pi\(and\)\left|x\right| \le \,1 \,\,\,\,\,\dots(i)$ 
and the RHS of given equation becomes 
\(sin^{-1} \left(2\,cos \,y \,sin \,y\right)=sin^{-1} \left(sin\, 2y\right)=2y\) 
Since, \(sin^{-1} \left(2x \sqrt{1-x^{2}}\right)\) lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}.\)
\(\therefore\) 2y lies between \(-\frac{\pi}{2}\) and \(\frac{\pi}{2}\). 
i.e., y lies between - \(\frac{\pi}{4}\) and \(\frac{\pi}{4}\). 
\(\therefore -\frac{\pi}{4}\le y\le\frac{\pi}{4}.\,\,\,\,\,\dots(ii)\) 
On combining Eqs. (i) and (ii), we get 
\(0\le\, y\le\,\frac{\pi}{4}\)
\(\Rightarrow 1\ge cos \,y \ge\frac{1}{\sqrt{2}}\)
\(\Rightarrow \frac{1}{\sqrt{2}} \le x \le 1\)
\(\Rightarrow x \in\left[\frac{1}{\sqrt{2}}, 1\right]\)

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