Calculating Trigonometric Expression Value
We are given $\cot A = \sqrt{5}$ and need to find the value of the expression $\frac{\csc^2 A - \sec^2 A}{\csc^2 A + \sec^2 A}$.
Finding Key Trigonometric Values
First, let's find the values of $\csc^2 A$ and $\sec^2 A$ using trigonometric identities.
- Using the identity $\csc^2 A = 1 + \cot^2 A$: $ \csc^2 A = 1 + (\sqrt{5})^2 = 1 + 5 = 6 $
- We know that $\tan A = \frac{1}{\cot A}$. So, $\tan A = \frac{1}{\sqrt{5}}$. Using the identity $\sec^2 A = 1 + \tan^2 A$: $ \sec^2 A = 1 + \left(\frac{1}{\sqrt{5}}\right)^2 = 1 + \frac{1}{5} = \frac{5+1}{5} = \frac{6}{5} $
Substituting Values into the Expression
Now, substitute the calculated values of $\csc^2 A$ and $\sec^2 A$ into the given expression:
$ \frac{\csc^2 A - \sec^2 A}{\csc^2 A + \sec^2 A} = \frac{6 - \frac{6}{5}}{6 + \frac{6}{5}} $Simplifying the Expression
Simplify the numerator and the denominator:
- Numerator: $ 6 - \frac{6}{5} = \frac{30 - 6}{5} = \frac{24}{5} $
- Denominator: $ 6 + \frac{6}{5} = \frac{30 + 6}{5} = \frac{36}{5} $
Now, divide the numerator by the denominator:
$ \frac{\frac{24}{5}}{\frac{36}{5}} = \frac{24}{5} \times \frac{5}{36} = \frac{24}{36} $Simplify the fraction:
$ \frac{24}{36} = \frac{2 \times 12}{3 \times 12} = \frac{2}{3} $Therefore, the value of the expression is $\frac{2}{3}$.
Similar Questions
- Find the value of $\frac{\cos^6 \theta + \sin^6 \theta + 3 \sin^2 \theta \cos^2 \theta}{\csc \theta \sec \theta ((\sin \theta + \cos \theta)^2 - 1)}$
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- If $\tan A = \frac{8}{15}$ then find $\sin A + \cos A$.
- If $\sin \theta - \cos \theta = 0$, and $0 \le \theta \le 90^\circ$, then the value of $\sin \theta + \cos \theta$ is:
- If $1 + \tan \theta = \sqrt{3}$, then $\sqrt{3} \cot \theta - 1 = ?$
- $\cos(x - y) = \frac{\sqrt{3}}{2}$, and $\sin(x + y) = 1$, where $x$ and $y$ are positive acute angles and $x \ge y$, then $x$ and $y$ are:
- Simplify: $\frac{\sin A \tan A}{(1 - \cos A)}$
Important Questions from Trigonometric Ratios and Identities
If cosec θ = 13/12, then sin θ + cos θ - tan θ is equal to:
What is the value of \(\frac{3 \sin 58^{\circ}}{\cos 32^{\circ}}+\frac{3 \sin 42^{\circ}}{\cos 48^{\circ}}\) ?
If \(\frac{{\tan \theta + \sin \theta }}{{\tan \theta - \sin \theta }} = \frac{{k + 1}}{{k - 1}},\) then k = ?
If α + β = 90° and α = 2β, then the value of 3 cos 2 α - 2 sin 2 β is equal to:
If \(\sqrt{3}\) tan θ = 3 sin θ, then what is the value of sin 2θ − cos 2θ ?
