Ravi Kant Kumar

Find α − β from inverse tangent expressions Question If \[ \alpha = \tan^{-1}\left(\frac{\sqrt{3}x}{2y – x}\right), \quad \beta = \tan^{-1}\left(\frac{2x – y}{\sqrt{3}y}\right) \] Find \( \alpha – \beta \). Solution Use identity: \[ \tan(\alpha – \beta) = \frac{\tan\alpha – \tan\beta}{1 + \tan\alpha \tan\beta} \] Substitute: \[ \tan(\alpha – \beta) = \frac{\frac{\sqrt{3}x}{2y – x} – \frac{2x […]

Find expression from cos⁻¹(x/3) + cos⁻¹(y/2) = θ/2 Question If \[ \cos^{-1}\left(\frac{x}{3}\right) + \cos^{-1}\left(\frac{y}{2}\right) = \frac{\theta}{2} \] Find: \[ 4x^2 – 12xy\cos\frac{\theta}{2} + 9y^2 \] Solution Let \[ \cos^{-1}\left(\frac{x}{3}\right) = A,\quad \cos^{-1}\left(\frac{y}{2}\right) = B \] Then, \[ A + B = \frac{\theta}{2} \] So, \[ \cos A = \frac{x}{3}, \quad \cos B = \frac{y}{2} \]

Find tan(π/4 − u/2) Question If \[ u = \cot^{-1}(\sqrt{\tan\theta}) – \tan^{-1}(\sqrt{\tan\theta}) \] Find: \[ \tan\left(\frac{\pi}{4} – \frac{u}{2}\right) \] Solution Let \[ x = \sqrt{\tan\theta} \] Then, \[ u = \cot^{-1}x – \tan^{-1}x \] Use identity: \[ \cot^{-1}x = \frac{\pi}{2} – \tan^{-1}x \] So, \[ u = \left(\frac{\pi}{2} – \tan^{-1}x\right) – \tan^{-1}x = \frac{\pi}{2} –

If x < 0, y < 0 and xy = 1, find tan⁻¹x + tan⁻¹y Question If \( x < 0 \), \( y < 0 \) and \( xy = 1 \), find: \[ \tan^{-1}x + \tan^{-1}y \] Solution We use identity: \[ \tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x+y}{1-xy}\right) \] Given: \[ xy = 1

Number of solutions of √(1+cos2x) = √2 sin⁻¹(sin x) Question Find the number of real solutions of: \[ \sqrt{1+\cos 2x} = \sqrt{2}\,\sin^{-1}(\sin x), \quad -\pi \le x \le \pi \] Solution Simplify LHS: \[ 1 + \cos 2x = 2\cos^2 x \] \[ \sqrt{1+\cos 2x} = \sqrt{2\cos^2 x} = \sqrt{2}\,|\cos x| \] So equation becomes:

Relation between α and β Question If \[ \alpha = \tan^{-1}(\tan \tfrac{5\pi}{4}), \quad \beta = \tan^{-1}(-\tan \tfrac{2\pi}{3}) \] Find the relation between \( \alpha \) and \( \beta \). Solution Principal value range of \( \tan^{-1}x \) is: \[ \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \] Find α: \[ \tan \tfrac{5\pi}{4} = 1 \Rightarrow \alpha = \tan^{-1}(1) = \frac{\pi}{4}

Number of solutions of tan⁻¹(2x) + tan⁻¹(3x) = π/4 Question Find the number of solutions of: \[ \tan^{-1}(2x) + \tan^{-1}(3x) = \frac{\pi}{4} \] Solution Use identity: \[ \tan^{-1}a + \tan^{-1}b = \tan^{-1}\left(\frac{a+b}{1-ab}\right) \] So, \[ \tan^{-1}\left(\frac{2x + 3x}{1 – 6x^2}\right) = \frac{\pi}{4} \] \[ \tan^{-1}\left(\frac{5x}{1 – 6x^2}\right) = \frac{\pi}{4} \] Thus, \[ \frac{5x}{1 – 6x^2}

Value of sin[cot⁻¹{tan(cos⁻¹x)}] Question Simplify: \[ \sin\left[\cot^{-1}\{\tan(\cos^{-1}x)\}\right] \] Solution Let \[ \cos^{-1}x = \theta \Rightarrow \cos\theta = x \] Then, \[ \tan(\cos^{-1}x) = \tan\theta \] Now, \[ \sin\left(\cot^{-1}(\tan\theta)\right) \] Let \[ \cot^{-1}(\tan\theta) = \phi \Rightarrow \cot\phi = \tan\theta \Rightarrow \tan\phi = \cot\theta \] So, \[ \phi = \frac{\pi}{2} – \theta \] Thus, \[ \sin\phi =

If sin⁻¹x − cos⁻¹x = π/6, find x Question If \[ \sin^{-1}x – \cos^{-1}x = \frac{\pi}{6} \] Find \( x \). Solution We use identity: \[ \sin^{-1}x + \cos^{-1}x = \frac{\pi}{2} \] Let \( \sin^{-1}x = \theta \) Then: \[ \theta – \left(\frac{\pi}{2} – \theta\right) = \frac{\pi}{6} \] \[ 2\theta – \frac{\pi}{2} = \frac{\pi}{6} \]

Positive integral solution of given inverse trig equation Question Find the positive integral solution of: \[ \tan^{-1}x + \cos^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right) = \sin^{-1}\left(\frac{3}{\sqrt{10}}\right) \] Solution We know identity: \[ \cos^{-1}\left(\frac{y}{\sqrt{1+y^2}}\right) = \tan^{-1}\left(\frac{1}{y}\right) \] So equation becomes: \[ \tan^{-1}x + \tan^{-1}\left(\frac{1}{y}\right) = \sin^{-1}\left(\frac{3}{\sqrt{10}}\right) \] Now, \[ \sin^{-1}\left(\frac{3}{\sqrt{10}}\right) = \tan^{-1}(3) \] (since \( \sin\theta = 3/\sqrt{10} \Rightarrow \tan\theta =