Why must x be acute for the identity $\sqrt{\frac{1-\mathrm{sin}x}{1+\mathrm{sin}x}}=\mathrm{sec}x-\mathrm{tan}x$ to hold true?

Tori Hines
2022-01-24
Answered

Why must x be acute for the identity $\sqrt{\frac{1-\mathrm{sin}x}{1+\mathrm{sin}x}}=\mathrm{sec}x-\mathrm{tan}x$ to hold true?

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izumrledk

Answered 2022-01-25
Author has **15** answers

The first quadrant is $0}^{\circ}\le x\le {90}^{\circ$ and the fourth quadrant is $270}^{\circ}\le x\le {360}^{\circ$ or this is the same as $-{90}^{\circ}\le x\le {0}^{\circ}$ . That is

$-{90}^{\circ}\le x\le {90}^{\circ}$

represents both the first and the fourth quadrant.

represents both the first and the fourth quadrant.

Serifluinueyk

Answered 2022-01-26
Author has **7** answers

The important point is that “identity” is true only when $-{90}^{\circ}<x<{90}^{\circ}$ .

In another word, it is not true when the terminal arm falls in QII and QIII.

Note that the angle lies in$-{90}^{\circ}<x<{0}^{\circ}$ is also an acute angle, because the minus sign only refers to moving the terminal arm in the clockwise direction. The resultant angle is still acute.

Added: To move the terminal arm through$330}^{\circ$ in the anticlockwise direction is the same as moving it through $30}^{\circ$ in the clockwise direction.

In another word, it is not true when the terminal arm falls in QII and QIII.

Note that the angle lies in

Added: To move the terminal arm through

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