Ravi Kant Kumar

Prove Advanced tan⁻¹ Identity Prove that for \(a,b,x,y > 0\), \[ \frac{2}{3}\tan^{-1}\left(\frac{3ab^2 – a^3}{b^3 – 3a^2b}\right) + \frac{2}{3}\tan^{-1}\left(\frac{3xy^2 – x^3}{y^3 – 3x^2y}\right) = \tan^{-1}\left(\frac{2\alpha\beta}{\alpha^2 – \beta^2}\right) \] where \( \alpha = -ax + by,\; \beta = bx + ay \) Solution: Use the identity: \[ \tan(3\theta) = \frac{3\tan\theta – \tan^3\theta}{1 – 3\tan^2\theta} \] This implies: […]

Prove tan⁻¹ Identity with α=ax−by, β=ay+bx Prove that \( \tan^{-1}\left(\frac{2ab}{a^2-b^2}\right) + \tan^{-1}\left(\frac{2xy}{x^2-y^2}\right) = \tan^{-1}\left(\frac{2\alpha\beta}{\alpha^2-\beta^2}\right) \) where \( \alpha = ax – by,\; \beta = ay + bx \) Solution: Use identity: \[ \tan^{-1}\left(\frac{2t}{1-t^2}\right) = 2\tan^{-1}(t) \] Rewrite: \[ \tan^{-1}\left(\frac{2ab}{a^2-b^2}\right) = 2\tan^{-1}\left(\frac{b}{a}\right) \] \[ \tan^{-1}\left(\frac{2xy}{x^2-y^2}\right) = 2\tan^{-1}\left(\frac{y}{x}\right) \] So LHS becomes: \[ 2\tan^{-1}\left(\frac{b}{a}\right) + 2\tan^{-1}\left(\frac{y}{x}\right) \]

Prove 2tan⁻¹{√(a−b)/√(a+b) tan(θ/2)} = cos⁻¹{(a cosθ + b)/(a + b cosθ)} Prove that \( 2\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan\frac{\theta}{2}\right) = \cos^{-1}\left(\frac{a\cos\theta + b}{a + b\cos\theta}\right) \) Solution: Let \[ t = \sqrt{\frac{a-b}{a+b}} \tan\frac{\theta}{2} \] Then LHS becomes: \[ 2\tan^{-1}(t) \] Use identity: \[ \cos(2\tan^{-1}t) = \frac{1 – t^2}{1 + t^2} \] So, \[ \cos(\text{LHS}) = \frac{1 – t^2}{1

Prove 2tan⁻¹((x−2)/(x−1)) + tan⁻¹((x+2)/(x+1)) = π/4 Prove that \( 2\tan^{-1}\left(\frac{x-2}{x-1}\right) + \tan^{-1}\left(\frac{x+2}{x+1}\right) = \frac{\pi}{4} \) Solution: Let \[ \theta = \tan^{-1}\left(\frac{x-2}{x-1}\right) \Rightarrow \tan\theta = \frac{x-2}{x-1} \] Using identity: \[ \tan(2\theta) = \frac{2\tan\theta}{1 – \tan^2\theta} \] \[ = \frac{2\cdot\frac{x-2}{x-1}}{1 – \left(\frac{x-2}{x-1}\right)^2} \] \[ = \frac{2(x-2)(x-1)}{(x-1)^2 – (x-2)^2} \] \[ = \frac{2(x-2)(x-1)}{(x^2 -2x +1) – (x^2 -4x

Solve cos⁻¹((x²−1)/(x²+1)) + ½tan⁻¹(2x/(1−x²)) Solve \( \cos^{-1}\left(\frac{x^2 – 1}{x^2 + 1}\right) + \frac{1}{2}\tan^{-1}\left(\frac{2x}{1 – x^2}\right) = \frac{2\pi}{3} \) Solution: Use identities: \[ \cos^{-1}\left(\frac{1 – t^2}{1 + t^2}\right) = 2\tan^{-1}(t) \] Here, \[ \frac{x^2 – 1}{x^2 + 1} = -\frac{1 – x^2}{1 + x^2} \] So, \[ \cos^{-1}\left(\frac{x^2 – 1}{x^2 + 1}\right) = \pi – 2\tan^{-1}(x)

Solve 2tan⁻¹(sin x) = tan⁻¹(2sec x) Solve \( 2\tan^{-1}(\sin x) = \tan^{-1}(2\sec x), \quad x \ne \frac{\pi}{2} \) Solution: Use identity: \[ \tan(2\theta) = \frac{2\tan\theta}{1 – \tan^2\theta} \] Let \[ \theta = \tan^{-1}(\sin x) \Rightarrow \tan \theta = \sin x \] Then, \[ \tan(2\theta) = \frac{2\sin x}{1 – \sin^2 x} = \frac{2\sin x}{\cos^2 x} =

Solve tan⁻¹(2x/(1−x²)) + cot⁻¹((1−x²)/2x) = 2π/3 Solve \( \tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{2\pi}{3}, \quad x > 0 \) Solution: Use identity: \[ \cot^{-1}(t) = \tan^{-1}\left(\frac{1}{t}\right) \] So, \[ \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \tan^{-1}\left(\frac{2x}{1-x^2}\right) \] Thus equation becomes: \[ 2\tan^{-1}\left(\frac{2x}{1-x^2}\right) = \frac{2\pi}{3} \] \[ \Rightarrow \tan^{-1}\left(\frac{2x}{1-x^2}\right) = \frac{\pi}{3} \] Taking tangent: \[ \frac{2x}{1-x^2} = \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

Solve 3sin⁻¹(2x/(1+x²)) − 4cos⁻¹((1−x²)/(1+x²)) + 2tan⁻¹(2x/(1−x²)) Solve \( 3\sin^{-1}\left(\frac{2x}{1+x^2}\right) – 4\cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) + 2\tan^{-1}\left(\frac{2x}{1-x^2}\right) = \frac{\pi}{3} \) Solution: Use standard identities: \[ \sin^{-1}\left(\frac{2x}{1+x^2}\right) = 2\tan^{-1}(x) \] \[ \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) = 2\tan^{-1}(x) \] \[ \tan^{-1}\left(\frac{2x}{1-x^2}\right) = 2\tan^{-1}(x) \] Substitute into equation: \[ 3(2\tan^{-1}x) – 4(2\tan^{-1}x) + 2(2\tan^{-1}x) = \frac{\pi}{3} \] \[ 6\tan^{-1}x – 8\tan^{-1}x + 4\tan^{-1}x = \frac{\pi}{3}

Solve tan⁻¹(1/4) + 2tan⁻¹(1/5) + tan⁻¹(1/6) + tan⁻¹(1/x) = π/4 Solve \( \tan^{-1}\left(\frac{1}{4}\right) + 2\tan^{-1}\left(\frac{1}{5}\right) + \tan^{-1}\left(\frac{1}{6}\right) + \tan^{-1}\left(\frac{1}{x}\right) = \frac{\pi}{4} \) Solution: Step 1: Evaluate \(2\tan^{-1}(1/5)\) \[ \tan(2\theta) = \frac{2\tan\theta}{1 – \tan^2\theta} \Rightarrow \tan(2\tan^{-1}(1/5)) = \frac{2/5}{1 – 1/25} = \frac{2/5}{24/25} = \frac{5}{12} \] \[ \Rightarrow 2\tan^{-1}\left(\frac{1}{5}\right) = \tan^{-1}\left(\frac{5}{12}\right) \] Step 2: Combine first two

Find cos(sec⁻¹x + cosec⁻¹x) Find the value of \( \cos\left(\sec^{-1}x + \csc^{-1}x\right), \quad |x| \ge 1 \) Solution: Let \[ \theta = \sec^{-1}x \Rightarrow \sec \theta = x \Rightarrow \cos \theta = \frac{1}{x} \] \[ \phi = \csc^{-1}x \Rightarrow \csc \phi = x \Rightarrow \sin \phi = \frac{1}{x} \] Now, \[ \sin \theta = \sqrt{1