April 2026

Solve 5tan⁻¹x + 3cot⁻¹x = 2π Problem Solve: \( 5\tan^{-1}x + 3\cot^{-1}x = 2\pi \) Solution Step 1: Use identity \[ \cot^{-1}x = \frac{\pi}{2} – \tan^{-1}x \] Step 2: Substitute \[ 5\tan^{-1}x + 3\left(\frac{\pi}{2} – \tan^{-1}x\right) = 2\pi \] \[ 5\tan^{-1}x + \frac{3\pi}{2} – 3\tan^{-1}x = 2\pi \] \[ 2\tan^{-1}x + \frac{3\pi}{2} = 2\pi \] […]

Solve tan⁻¹x + 2cot⁻¹x = 2π/3 Problem Solve: \( \tan^{-1}x + 2\cot^{-1}x = \frac{2\pi}{3} \) Solution Step 1: Use identity \[ \cot^{-1}x = \frac{\pi}{2} – \tan^{-1}x \] Step 2: Substitute \[ \tan^{-1}x + 2\left(\frac{\pi}{2} – \tan^{-1}x\right) = \frac{2\pi}{3} \] \[ \tan^{-1}x + \pi – 2\tan^{-1}x = \frac{2\pi}{3} \] \[ \pi – \tan^{-1}x = \frac{2\pi}{3} \]

Solve 4sin⁻¹x = π − cos⁻¹x Problem Solve: \( 4\sin^{-1}x = \pi – \cos^{-1}x \) Solution Step 1: Use identity \[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}x \] Step 2: Substitute \[ 4\sin^{-1}x = \pi – \left(\frac{\pi}{2} – \sin^{-1}x\right) \] \[ 4\sin^{-1}x = \frac{\pi}{2} + \sin^{-1}x \] Step 3: Solve \[ 3\sin^{-1}x = \frac{\pi}{2} \] \[

Solve sin⁻¹x = π/6 + cos⁻¹x Problem Solve: \( \sin^{-1}x = \frac{\pi}{6} + \cos^{-1}x \) Solution Step 1: Use identity \[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}x \] Step 2: Substitute \[ \sin^{-1}x = \frac{\pi}{6} + \left(\frac{\pi}{2} – \sin^{-1}x\right) \] \[ \sin^{-1}x = \frac{2\pi}{3} – \sin^{-1}x \] Step 3: Solve \[ 2\sin^{-1}x = \frac{2\pi}{3} \] \[

Solve sin(sin⁻¹(1/5) + cos⁻¹x) = 1 Problem Solve: \( \sin\left(\sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}(x)\right) = 1 \) Solution Step 1: Use property of sine \[ \sin \theta = 1 \Rightarrow \theta = \frac{\pi}{2} \] So, \[ \sin^{-1}\left(\frac{1}{5}\right) + \cos^{-1}(x) = \frac{\pi}{2} \] Step 2: Convert using identity \[ \cos^{-1}x = \frac{\pi}{2} – \sin^{-1}x \] Step 3: Substitute

Find x in (sin⁻¹x)² + (cos⁻¹x)² = 17π²/36 Problem Solve: \( (\sin^{-1}x)^2 + (\cos^{-1}x)^2 = \frac{17\pi^2}{36} \) Solution Step 1: Use identity \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \] Let \( a = \sin^{-1}x \), then: \[ \cos^{-1}x = \frac{\pi}{2} – a \] Step 2: Substitute \[ a^2 + \left(\frac{\pi}{2} – a\right)^2 = \frac{17\pi^2}{36} \]

Find x in cot(cos⁻¹(3/5) + sin⁻¹(x)) = 0 Problem Find x if \( \cot\left(\cos^{-1}\left(\frac{3}{5}\right) + \sin^{-1}(x)\right) = 0 \) Solution Step 1: Use property of cot \[ \cot \theta = 0 \Rightarrow \theta = \frac{\pi}{2} \] So, \[ \cos^{-1}\left(\frac{3}{5}\right) + \sin^{-1}(x) = \frac{\pi}{2} \] Step 2: Convert using identity \[ \cos^{-1}t = \frac{\pi}{2} – \sin^{-1}t

Find x and y Problem If \( \sin^{-1}(x) + \sin^{-1}(y) = \frac{\pi}{3} \) and \( \cos^{-1}(x) – \cos^{-1}(y) = \frac{\pi}{6} \), find x and y. Solution Step 1: Convert cos⁻¹ into sin⁻¹ \[ \cos^{-1}t = \frac{\pi}{2} – \sin^{-1}t \] So, \[ \cos^{-1}x – \cos^{-1}y = \left(\frac{\pi}{2} – \sin^{-1}x\right) – \left(\frac{\pi}{2} – \sin^{-1}y\right) \] \[ =

Find sin⁻¹x + sin⁻¹y Problem If \( \cos^{-1}x + \cos^{-1}y = \frac{\pi}{4} \), find \( \sin^{-1}x + \sin^{-1}y \). Solution Step 1: Use identity \[ \sin^{-1}t + \cos^{-1}t = \frac{\pi}{2} \] So, \[ \sin^{-1}x = \frac{\pi}{2} – \cos^{-1}x \] \[ \sin^{-1}y = \frac{\pi}{2} – \cos^{-1}y \] Step 2: Add both \[ \sin^{-1}x + \sin^{-1}y =

Evaluate cos(sec⁻¹x + cosec⁻¹x) Problem Evaluate: \( \cos\left(\sec^{-1}x + \csc^{-1}x\right), \quad |x| \ge 1 \) Solution Let: \[ A = \sec^{-1}x,\quad B = \csc^{-1}x \] Step 1: Convert to sine and cosine \[ \sec A = x \Rightarrow \cos A = \frac{1}{x} \] \[ \csc B = x \Rightarrow \sin B = \frac{1}{x} \] Step