Find the value of \(\frac{\tan A}{1 - \sqrt{\mathrm{cosec}^2 A - 1}} + \frac{\cot A}{1 - \sqrt{\sec^2 A - 1}}\).
\(1 + \tan A + \cot A\)
Use the identities \(\mathrm{cosec}^2 A - 1 = \cot^2 A\) and \(\sec^2 A - 1 = \tan^2 A\), so the roots become \(\cot A\) and \(\tan A\).
The expression becomes \(\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}\).
Let \(\tan A = t\), so \(\cot A = \frac{1}{t}\). Then \(\frac{t}{1 - \frac{1}{t}} = \frac{t^2}{t - 1}\) and \(\frac{\frac{1}{t}}{1 - t} = -\frac{1}{t(t - 1)}\).
Adding: \(\frac{t^3 - 1}{t(t - 1)} = \frac{(t - 1)(t^2 + t + 1)}{t(t - 1)} = \frac{t^2 + t + 1}{t}\).
\(\frac{t^2 + t + 1}{t} = t + 1 + \frac{1}{t} = 1 + \tan A + \cot A\).
Hence, the answer is \(1 + \tan A + \cot A\).
Similar Questions
Simplify: \(\frac{2\sin^2A-\sin^4A+\cos^4A}{3\sin^4A-2\sin^6A+3\cos^4A-2\cos^6A}\)
The value of \(\frac{\sin\theta}{\sec\theta+\tan\theta-1}+\frac{\cos\theta}{\operatorname{cosec}\theta+\cot\theta-1}\) is
If \(\frac{1 + \cos A}{1 - \cos A} = \left(\frac{m}{n}\right)^2\), then find the value of \(\operatorname{cosec} A - \cot A\).
Find the value of \(\sqrt{\frac{\sec^2\theta + \mathrm{cosec}^2\theta}{\tan^2\theta - \sin^2\theta}}\).
If θ lies in the first quadrant and \(\sec^2\theta + \cos^2\theta = 2\), then the value of \(\frac{14\cos\theta}{5\cos^2\theta - 3\cos\theta + 5}\) is:
Find the value of \(\frac{2 + \tan^2\theta + \cot^2\theta}{\sec\theta\,\mathrm{cosec}\,\theta}\).
If \(\frac{1}{(\csc\theta+1)}+\frac{1}{(\csc\theta-1)}=2\sec\theta,\ 0^\circ<\theta<90^\circ\), then the value of \(\frac{\cot\theta-2\csc\theta}{\sec\theta}\) is:
If \((\cos\theta + \sin\theta) : (\cos\theta - \sin\theta) = (\sqrt{3} + 1) : (\sqrt{3} - 1), 0^\circ \lt \theta \lt 90^\circ\), then the value of \(\frac{\tan(\theta + 30^{\circ})}{\sec(\theta - 30^{\circ})}\) is:
If \(a \sin^{3}\theta + b \cos^{3}\theta = \sin\theta \cos\theta\) and \(a \sin\theta = b \cos\theta\), then value of \(a^{2} + b^{2}\) is
Find the value of \(\frac{\sin \theta - \cos \theta + 1}{\sin \theta + \cos \theta - 1}\).
Important Questions from Trigonometric Ratios and Identities
What is the ratio of the greatest to the smallest value of 2 – 2 sin x – sin 2x, 0 ≤ x ≤ (π/2)?
If sinθ = \(\frac{4}{5}\) , Find the value of sin3θ
If x, y are acute angles, where 0 < x + y < 90° and sin(3x - 40°) = cos (3y + 40°), then the value of tan (x + y) is equal to
What is (1 + cot θ - cosec θ)(1 + tan θ + sec θ) equal to?
If cos x = p/q and 0° < x < 90°, then the value of tan x is:
