Question

If cot A = `sqrt5`, the value of cosec² A − sec² A is ______.

Options

• `5/24`

• `4 4/5`

• 5

• 24

Answer 1

If cot A = `sqrt5`, the value of cosec² A − sec² A is `bbunderline(4 4/5)`.

Explanation:

Using standard trigonometric identities:

• \[\csc^2 A = 1 + \cot^2 A = 1 + (\sqrt{5})^2 = 1 + 5 = 6\]
• \[\tan A = \frac{1}{\cot A} = \frac{1}{\sqrt{5}}\]
• \[\sec^2 A = 1 + \tan^2 A = 1 + \left(\frac{1}{\sqrt{5}}\right)^2 = 1 + \frac{1}{5} = \frac{6}{5}\]

Subtracting the two values gives:

\[\csc^2 A - \sec^2 A = 6 - \frac{6}{5} = \frac{30 - 6}{5} = \frac{24}{5} = 4\frac{4}{5}\]