Why must x be acute for the identity 1−sin⁡x1+sin⁡x=sec⁡x−tan⁡x to hold true?

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izumrledk · Accepted Answer

The first quadrant is 0∘≤x≤90∘ and the fourth quadrant is 270∘≤x≤360∘ or this is the same as −90∘≤x≤0∘. That is  −90∘≤x≤90∘  represents both the first and the fourth quadrant.

Serifluinueyk · Accepted Answer

The important point is that “identity” is true only when −90∘&amp;lt;x&amp;lt;90∘.  In another word, it is not true when the terminal arm falls in QII and QIII.  Note that the angle lies in −90∘&amp;lt;x&amp;lt;0∘ 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∘ in the anticlockwise direction is the same as moving it through 30∘ in the clockwise direction.

Why must x be acute for the identity \sqrt{\frac{1-\sin x}{1+\sin

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