\({\left( {\frac{{1 - tan\theta }}{{1 - \cot \theta }}} \right)^2} + 1 = ?\)
sec 2θ
Step 1 — Simplify the inner fraction: write \(\tan\theta=\tfrac{\sin\theta}{\cos\theta}\) and \(\cot\theta=\tfrac{\cos\theta}{\sin\theta}\).
\[\frac{1-\tan\theta}{1-\cot\theta}=\frac{\frac{\cos\theta-\sin\theta}{\cos\theta}}{\frac{\sin\theta-\cos\theta}{\sin\theta}}=\frac{\sin\theta}{\cos\theta}\cdot\frac{\cos\theta-\sin\theta}{-(\cos\theta-\sin\theta)}=-\tan\theta\]
Step 2 — Square and add 1:
\[(-\tan\theta)^2+1=\tan^2\theta+1=\sec^2\theta\]
Therefore the value is \(\sec^2\theta\).
Similar Questions
Evaluate the following.
sin 25° sin 65° – cos 25° cos 65°.
If sin 23° = \(\frac{a}{b}\), then the value of sec 23° - sin 67° is __________.
If sec2 θ + tan2 θ = \(\frac{25}{18}\), then the value of sec4 θ - tan4 θ is:
If \(\frac{\cos \beta}{\sec \alpha}\) = 15 and \(\frac{\sin \beta}{\sec \alpha}\) = 16, then the value of sin2β is ___________.
If sec θ + tan θ = 5, (θ ≠ 0), then sec θ is equal to:
If sec θ + cos θ = 32, then sec2 θ+ cos2 θ is ______.
If sin t + cos t = \(\frac{4}{5}\), then find sin t. cos t.
Select the INCORRECT formula from the following options.
If Sin θ = \(\frac{4}{5}\), find the value of tan θ - Cot θ.
If a cot θ + b cosec θ = p and b cot θ + a cosec θ = q then p2 - q2 is equal to ______.
Important Questions from Trigonometric Ratios and Identities
The value of 4 sin 230° + 3 cot 260° - 2 tan 245° is:
The value of 1 - sin 35° cos 55° is equal to:
If sin 3 θ = cos ( θ – 6°), then θ is:
If θ = 45°, then what will be the value of \(\frac{{\\sin \,\theta \, + \,\cos \,\theta }}{{\sin \,\theta \, - \,\cos \,\theta }}\) ?
If sin A = \(\frac{1}{2}\) and cos B = \(\frac{1}{2}\) then find A + B.
