If \(\cos A = \frac{4}{5}\), find the value of \(\dfrac{\sin^2 A - 5\cos^2 A + \sec^2 A}{\tan^2 A}\).
\(\frac{-511}{225}\)
Given \(\cos A = \frac{4}{5}\), so \(\sin A = \sqrt{1-\frac{16}{25}} = \frac{3}{5}\).
Then \(\tan A = \frac{\sin A}{\cos A} = \frac{3}{4}\) and \(\sec A = \frac{1}{\cos A} = \frac{5}{4}\).
Numerator \(= \sin^2 A - 5\cos^2 A + \sec^2 A = \frac{9}{25} - 5\cdot\frac{16}{25} + \frac{25}{16}\).
\(= \frac{9}{25} - \frac{80}{25} + \frac{25}{16} = -\frac{71}{25} + \frac{25}{16} = \frac{-1136 + 625}{400} = -\frac{511}{400}\).
Denominator \(= \tan^2 A = \frac{9}{16}\).
\(\dfrac{-\frac{511}{400}}{\frac{9}{16}} = -\frac{511}{400} \times \frac{16}{9} = -\frac{511}{225}\).
Hence, the value is \(-\frac{511}{225}\).
Similar Questions
- If $\cos(x + y) = \frac{1}{2}$ and $\sin(x - y) = 0$, where x and y are positive acute angles and $x \ge y$, then x and y are:
- If $P = \tan\theta + \sec\theta$, then the value of $\tan\theta$ is:
- If $\tan A = \frac{3}{4}$, then the value of $\frac{\cos A - \sin A}{\cos A + \sin A}$ is:
- What is the value of $\sin(48^\circ + \theta) - \cos(42^\circ - \theta)$?
- What is the value of the following expression?
$(\tan 2^\circ \tan 88^\circ) (\tan 3^\circ \tan 87^\circ) ... (\tan 43^\circ \tan 47^\circ) \tan 45^\circ$ - If the ratio of the sine of an acute angle to its cosine is 5 : 12, then what will the value of sine of that angle be?
- If $\tan\theta + \cot\theta = 6$, then find the value of $\tan^2\theta + \cot^2\theta$
- What is the value of the following expression?
$\frac{\cos 3x + \cos x}{\sin 3x - \sin x}$ - If $\sin(A - B) = \frac{1}{2}$ and $\cos(A + B) = \frac{1}{2}$ with $0^\circ < (A + B) \leq 90^\circ$, $A > B$, then find the measure of A and B.
- If $\sqrt{2} \sin(5x - 5)^\circ = \tan 45^\circ$, then the value of x (in degrees) is:
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 \(\sin \theta =\frac{3}{5}\) and \(\cos \theta =\frac{4}{5}\) , then the value of \(\frac{1+\tan \theta}{1-\cot \theta}\) is:
