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Chapter 5: Problem 41

Verify the identity. $$\sqrt{\frac{1+\sin \theta}{1-\sin \theta}}=\frac{1+\sin \theta}{|\cos \theta|}$$

Short Answer

Expert verified
The identity \(\sqrt {\frac {1 + \sin \theta}{1 - \sin \theta}} = \frac {1 + \sin \theta}{|\cos \theta|}\) is verified.

Step by step solution

01

Simplify the Left Hand Side

Begin by simplifying the left-hand side (LHS) of the equation. The LHS represents the square root of a fraction, and this can be simplified by taking the square root of the numerator and the square root of the denominator separately, we get: \[ \sqrt {\frac {1 + \sin \theta}{1 - \sin \theta}} = \frac {\sqrt {1 + \sin \theta}}{\sqrt {1 - \sin \theta}} \]
02

Use Trigonometric Identity to Simplify Denominator

The denominator \(\sqrt {1 - \sin \theta}\) can be simplified using the Pythagorean identity (\(\sin^2 \theta + \cos^2 \theta = 1\)). This gives us a new expression for the denominator in terms of \(\cos \theta\), we get: \[ \sqrt {1 - \sin \theta} = |\cos \theta| \] Therefore, the LHS of the equation now becomes: \[ \frac {\sqrt {1 + \sin \theta}}{|\cos \theta|} \]
03

Simplify the Numerator

For \(\theta\) in the range 0 to \( \pi \), \(\sin \theta \geq 0\), and the square root therefore does not change the sign of its argument. Thus, for such \(\theta\), we may write : \[ \sqrt{1 + \sin \theta} = 1 + \sin \theta \] Thus, the simplified LHS now becomes: \[ \frac {1 + \sin \theta}{|\cos \theta|} \] Which is the right-hand side of the equation, and this confirms the given identity.

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