Answer: Choice H) 2
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Explanation:
Recall that the pythagorean trig identity is [tex]\sin^2 x + \cos^2x = 1[/tex]
If we were to isolate sine, then,
[tex]\sin^2 x + \cos^2x = 1\\\\\sin^2 x = 1-\cos^2x\\\\\sin x = \sqrt{1-\cos^2x}\\\\[/tex]
We don't have to worry about the plus minus because sine is positive when 0 < x < pi/2.
Through similar calculations, [tex]\cos x = \sqrt{1-\sin^2x}\\\\[/tex]
Cosine is also positive in this quadrant.
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So,
[tex]\frac{\sqrt{1-\cos^2x}}{\sin x}+\frac{\sqrt{1-\sin^2x}}{\cos x}\\\\\frac{\sin x}{\sin x}+\frac{\cos x}{\cos x}\\\\1+1\\\\2[/tex]
Therefore,
[tex]\frac{\sqrt{1-\cos^2x}}{\sin x}+\frac{\sqrt{1-\sin^2x}}{\cos x}=2[/tex]
is an identity as long as 0 < x < pi/2
