Properties of inverse trigonometric functions

By definition, \(\sin^{-1}x\) is the principal angle \(y\in[-\tfrac{\pi}{2},\tfrac{\pi}{2}]\) such that \(\sin y=x\). Hence \(\sin(\sin^{-1}x)=x\) for all \(x\in[-1,1]\). The distractor "-x" is incorrect because taking \(\sin^{-1}\) returns the angle whose sine is exactly \(x\), not its negative. Exam tip: always check the domain \([-1,1]\) of \(\sin^{-1}x\) and remember the principal value range of arcsin when composing functions.

The function \(\cos^{-1}x\) (principal value in \([0,\pi]\)) returns an angle whose cosine is \(x\). Applying \(\cos\) to that angle gives back the original \(x\) for every \(x\in[-1,1]\). The option \(|x|\) is a tempting distractor (students may wrongly think the result must be nonnegative), but for negative \(x\) the angle \(\cos^{-1}x\) lies in \((\pi/2,\pi]\) and its cosine is negative, equal to \(x\). Exam tip: always check the domain and principal range of inverse trig functions before simplifying compositions.

The angle \(\tan^{-1}x\) (principal value of arctan) has tangent equal to \(x\). Since arctan takes values in \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\) where \(\tan\) is one-to-one, the composition gives \(\tan(\tan^{-1}x)=x\) for all real \(x\). Option B (0) holds only when \(x=0\); option C (\(1/x\)) and D (\(-x\)) are true only for special values and are not generally correct. Exam tip: remember the principal range of \(\tan^{-1}x\) so you can use the identity \(f(f^{-1}(x))=x\) when \(f\) is one-to-one on that range.

The principal value of \(\tan^{-1}x\) lies in \(\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\). The endpoints \(\pm\frac{\pi}{2}\) are excluded because tangent is undefined there. Option C is close, but it incorrectly includes these endpoints. Exam tip: Remember that the principal-value intervals for \(\tan^{-1}x\) and \(\cot^{-1}x\) are different.

The identity \(\sin^{-1}x+\cos^{-1}x=\frac{\pi}{2}\) applies because \(\frac{3}{5}\in\left[-1,1\right]\). Use the identity directly in such questions.

Use the identity \(\sin^{-1}x+\cos^{-1}x=\dfrac{\pi}{2}\) valid for \(x\in[-1,1]\). Here \(x=4/5\) lies in the domain, so the sum equals \(\dfrac{\pi}{2}\). Why other options fail: \(\frac{\pi}{6}\) (30°) is not the complement of the corresponding angles here, and \(\frac{3\pi}{2}\) or 0 are clearly inconsistent with principal values of arcsin/arccos. Exam tip: memorize the complementary identity and always check the domain before applying it.

When \(x>0\), \(tan^{-1}x+\tan^{-1}\left(\frac{1}{x}\right)=\frac{\pi}{2}\). Always check the sign condition.

Here \(2>0\), so \(tan^{-1}2+\tan^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{2}\). Identify (x) and \(\frac{1}{x}\) in the identity.

The identity \(\tan^{-1}x+\cot^{-1}x=\frac{\pi}{2}\) holds. For negative \(x\), the principal value of \(\cot^{-1}x\) is still taken in \(\left(0,\pi\right)\).

The function \(\sin^{-1}x\) (arcsin) is odd, meaning \(\sin^{-1}(-x)=-\sin^{-1}x\) for all \(x\in[-1,1]\). Hence the correct choice is \(-\sin^{-1}x\). Option B is incorrect because that would require the function to be even. Options C and D are related identities (\(\cos^{-1}x=\tfrac{\pi}{2}-\sin^{-1}x\)) but do not equal \(\sin^{-1}(-x)\). Exam tip: check odd/even properties of inverse trig functions first — they often give the quickest justification.

The arctan function is odd and has principal value range \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\). Hence for any real \(x\), \(\tan^{-1}(-x)= -\tan^{-1}x\). Option B is incorrect because it does not change the sign; options C and D do not generally equal \(\tan^{-1}(-x)\) within the principal range. Exam tip: check whether the inverse trig function is odd or even and remember its principal value interval.

From \(\tan\theta=\frac{3}{4}\), take opposite side (3) and adjacent side (4). The hypotenuse is (5), so \(\sin\theta=\frac{3}{5}\).

If \(\theta=\cos^{-1}x\), then \(\cos\theta=x\) and \(\theta\in\left[0,\pi\right]\). In this range \(\sin\theta\ge0\), so the value is \(\sqrt{1-x^2}\).

If \(\theta=\sin^{-1}x\), then \(\sin\theta=x\) and \(\theta\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]\). Here \(cos\theta\ge0\), so \(\sqrt{1-x^2}\) is correct.

If \(\theta=\sin^{-1}\left(\frac{3}{5}\right)\), then \(\sin\theta=\frac{3}{5}\) and \(\cos\theta=\frac{4}{5}\). Hence \(\tan\theta=\frac{3}{4}\).

If \(\theta=\cos^{-1}\left(\frac{5}{13}\right)\), then \(\cos\theta=\frac{5}{13}\) and \(\sin\theta=\frac{12}{13}\). Hence \(\cot\theta=\frac{5}{12}\).

Let \(\theta=\tan^{-1}(7/24)\), so \(\tan\theta=7/24=\) opposite/adjacent. Using a right triangle with opposite = 7 and adjacent = 24 gives hypotenuse = \(\sqrt{7^2+24^2}=25\). Then \(\sec\theta=1/\cos\theta=\) hypotenuse/adjacent = \(25/24\), so option A is correct. Closest distractor \(24/25\) (B) is the reciprocal and corresponds to \(\cos\theta\), not \(\sec\theta\). Option C (\(7/25\)) equals \(\sin\theta\), and D (\(25/7\)) equals \(\csc\theta\). Exam tip: draw the 7‑24‑25 right triangle for arctan problems and read off opposite, adjacent, hypotenuse to compute sin, cos, sec quickly.

If \(cos\theta=\frac{8}{17}\), then \(sin\theta=\frac{15}{17}\). Hence \(cosec\theta=\frac{17}{15}\).

Since \(\cos\frac{\pi}{3}=\frac{1}{2}\) and \(\sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}\). First find the inner value.

Since \(\sin\frac{\pi}{6}=\frac{1}{2}\) and \(\cos^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}\). Remember standard angles.

Since \(\cot\frac{\pi}{3}=\frac{1}{\sqrt{3}}\) and \(\tan^{-1}\left(\frac{1}{\sqrt{3}}\right)=\frac{\pi}{6}\). In such questions, find the trigonometric value first.

By definition, \(\sin^{-1}x\) gives the angle \(y\in[-\pi/2,\pi/2]\) such that \(\sin y=x\). Applying sine to that angle returns the original value: \(\sin(\sin^{-1}x)=x\) for \(x\in[-1,1]\). Option B (\(\sin^{-1}x\)) is incorrect because it is the inverse value itself; composing in the given order returns \(x\), not the inverse. Exam tip: always check the domain of the inner function and the principal range of the inverse to verify such compositions quickly.

The function \(\cos^{-1}x\) returns the principal angle \(\theta\in[0,\pi]\) such that \(\cos\theta=x\). Therefore \(\cos(\cos^{-1}x)=x\) for all \(x\in[-1,1]\). Option B (|x|) is incorrect because \(x\) may be negative and composition does not take absolute value. Option C (\(\cos^{-1}x\)) is an angle, not the original numeric value \(x\). Option D (\(-x\)) is also incorrect. Exam tip: remember domain/range restrictions — \(\cos(\cos^{-1}x)=x\) on \([-1,1]\), whereas \(\cos^{-1}(\cos y)\) equals \(y\) only when \(y\in[0,\pi]\).

\(\tan^{-1}x\) denotes the angle whose tangent is x. If \(\tan^{-1}x=0\), taking tan of both sides gives \(x=\tan0=0\). Also note arctan's principal value lies in \((-\tfrac{\pi}{2},\tfrac{\pi}{2})\), so the solution is unique. Option B (1) would be correct only if \(\tan^{-1}x=\tfrac{\pi}{4}\). Option D is an angle (\(\tfrac{\pi}{4}\)), not the value of x. Exam tip: clearly distinguish the inverse-trig function's output (an angle) from the variable x before applying the direct trig function.