For x such that 0 < x < (pi)/(2), the expression (sqrt(1-cos^(2)x))/(sin x)+(sqrt(1-sin^(2)x))/(cos x) is equivalent to: F. 0 G. 1 H. 2 J. -tan x K. sin 2x

Recognize Trigonometric Identities: Recognize that the expressions under the square roots are trigonometric identities. The identity sin^2(x) + cos^2(x) = 1 can be rearranged to 1 - cos^2(x) = sin^2(x) and 1 - sin^2(x) = cos^2(x).Substitute Identities: Substitute the identities into the original expression to simplify it. This gives us (sqrt(sin^2(x)))/(sin x) + (sqrt(cos^2(x)))/(cos x).Simplify Square Roots: Since we are given that 0 < x < (pi)/(2), both sin x and cos x are positive in this range. Therefore, we can simplify sqrt(sin^2(x)) to sin x and sqrt(cos^2(x)) to cos x.Combine Terms: After simplification, the expression becomes sin x / sin x + cos x / cos x, which simplifies to 1 + 1.Final Result: Adding the two terms together, we get 1 + 1 = 2.

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Q. For

x

such that

0<x<\frac{\pi}{2}

, the expression

\frac{\sqrt{1-\cos ^{2} x}}{\sin x}+\frac{\sqrt{1-\sin ^{2} x}}{\cos x}

is equivalent to:

\newline

0

\newline

1

\newline

2

\newline

-\tan x

\newline

\sin 2 x

Recognize Trigonometric Identities: Recognize that the expressions under the square roots are trigonometric identities. The identity $\sin^2(x) + \cos^2(x) = 1$ can be rearranged to $1 - \cos^2(x) = \sin^2(x)$ and $1 - \sin^2(x) = \cos^2(x)$ .
Substitute Identities: Substitute the identities into the original expression to simplify it. This gives us $(\sqrt{\sin^2(x)})/(\sin x) + (\sqrt{\cos^2(x)})/(\cos x)$ .
Simplify Square Roots: Since we are given that 0 < x < \frac{\pi}{2}, both $\sin x$ and $\cos x$ are positive in this range. Therefore, we can simplify $\sqrt{\sin^2(x)}$ to $\sin x$ and $\sqrt{\cos^2(x)}$ to $\cos x$ .
Combine Terms: After simplification, the expression becomes $\frac{\sin x}{\sin x} + \frac{\cos x}{\cos x}$ , which simplifies to $1 + 1$ .
Final Result: Adding the two terms together, we get $1 + 1 = 2$ .