First slide

Trigonometric Identities

Question

The value of $f (α) = \sqrt{{cosec}^{2} α - 2 \cot α} + \sqrt{{cosec}^{2} α + 2 \cot α}$ can be

Moderate

A

$2 \cot α$
B

$- 2 \cot α$
C

2
D

-2

Solution

$\begin{matrix} f (α) = \sqrt{{cosec}^{2} α - 2 \cot α} + \sqrt{{cosec}^{2} α + 2 \cot α} \\ = \sqrt{1 + \cot^{2} α - 2 \cot α} + \sqrt{1 + \cot^{2} α + 2 \cot α} \\ = | \cot α - 1 | + | \cot α + 1 | \end{matrix}$
$\begin{array}{l} CaseI : \cot α \leq - 1 \\ ∴ f (α) - - \cot α + 1 - \cot α - 1 = - 2 \cot α \\ CaseII : - 1 \leq \cot α \leq 1 \\ ∴ f (α) = - \cot α + 1 + \cot α + 1 = 2 \\ CaseIII : \cot α \geq 1 \\ ∴ f (α) = (\cot α - 1) + (\cot α + 1) = 2 \cot α \end{array}$

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