If A is an acute angle, what is the value of the following expression? \(\frac{1 + \cos A}{1 - \cos A} - 2\cot A \operatorname{cosec} A\)
\(\operatorname{cosec}^{2} A + \cot^{2} A\)
Given expression:
\(\frac{1 + \cos A}{1 - \cos A} - 2\cot A \operatorname{cosec} A\)
We know that:
\(2\cot A \operatorname{cosec} A = 2 \cdot \frac{\cos A}{\sin A} \cdot \frac{1}{\sin A} = \frac{2\cos A}{\sin^2 A} = \frac{2\cos A}{(1 - \cos A)(1 + \cos A)}\)
Expressing the first term with the common denominator:
\(\frac{1 + \cos A}{1 - \cos A} = \frac{(1 + \cos A)^2}{(1 - \cos A)(1 + \cos A)} = \frac{1 + 2\cos A + \cos^2 A}{\sin^2 A}\)
Subtracting the second term:
\(\frac{1 + 2\cos A + \cos^2 A - 2\cos A}{\sin^2 A} = \frac{1 + \cos^2 A}{\sin^2 A} = \frac{1}{\sin^2 A} + \frac{\cos^2 A}{\sin^2 A} = \operatorname{cosec}^2 A + \cot^2 A\)
Hence, the correct answer is \(\operatorname{cosec}^{2} A + \cot^{2} A\).
Similar Questions
The value of sin242° + sin248° + sec230° − cot245° is:
What is the value of [(1 + cot2A) + (1 + \(\frac{1}{\cot^2 A}\))] × sin2A cos2A?
If cosA = \(\frac{9}{41}\), A is an acute angle, then find the value of cosecA cosA + secA sinA.
If \(\sin\theta + \cos\theta = \sqrt{3}\), then the value of \(\sin^{3}\theta + \cos^{3}\theta\) is
The value of \(\tan^{2}\theta + \cot^{2}\theta - \sec^{2}\theta\cdot\csc^{2}\theta + 2\) is:
If \(\sec^{2}\alpha = \frac{4ab}{(a+b)^{2}}\), then:
What is the value of the following expression?
\(\tan A\left[\frac{\sec A + \tan A}{\operatorname{cosec}A - \cot A}\right]\)
If sinA = \(\frac{4}{5}\), A is an acute angle, then find the value of secA × sinA − cosecA × cosA.
The value of \(\left[\frac{\sin^{2}25^\circ + \sin^{2}65^\circ}{\cos^{2}25^\circ + \cos^{2}65^\circ} + \sin^{2}57^\circ + \cos 57^\circ \sin 33^\circ\right]\) is:
The value of (1 + sec 15° + cot 75°)(1 – cosec 15° + tan 75°) is equal to:
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:
