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Chapter 5: Problem 30

Verify the identity. $$\frac{\cos \theta \cot \theta}{1-\sin \theta}-1=\csc \theta$$

Short Answer

Expert verified
Yes, the given identity \(\frac{\cos \theta \cot \theta}{1-\sin \theta}-1=\csc \theta\) is correct.

Step by step solution

01

Convert trigonometric terms

The trigonometric identities should be converted to the equivalent expressions in terms of sine and cosine. So, \[\cot \theta\] becomes \(\frac{\cos \theta}{\sin \theta}\), and \[\csc \theta\] becomes \(\frac{1}{\sin \theta}\). Thus, the equation transforms into: \[\frac{\cos \theta \times \frac{\cos \theta}{\sin \theta}}{1-\sin \theta}-1=\frac{1}{\sin \theta}\].
02

Simplify Left Side

Now, simplify the left hand side (LHS) of the equation. The denominator and the numerator of the fraction part are multiplied by \(\sin \theta\) in order to get rid of the fraction in the numerator. This gives \[\frac{\cos^2 \theta}{\sin \theta-\sin^2 \theta}-1\]. After simplifying, it becomes: \[\frac{1 - \sin^2 \theta}{\sin \theta - \sin^2 \theta} - 1\]. Note that \(\cos^2 \theta\) is rewritten as \(1 - \sin^2 \theta\), using the Pythagorean identity.
03

Simplify Further

Notice that \(\sin \theta - \sin^2 \theta\) in the denominator can factored as \(\sin \theta (1 - \sin \theta)\). The fraction now becomes: \[\frac{1 - \sin^2 \theta}{\sin \theta (1- \sin \theta)} - 1\]. The conjugate in the numerator and denominator cancels out, giving: \[\frac{1}{\sin \theta} - 1\]
04

Compare

Now, compare this result to the right hand side (RHS) of the equation \[\frac{1}{\sin \theta}\] which is identical. Thus, the identity is verified.

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