Simplify: $$\frac{1}{(sec θ - tan θ)}$$  = ? ​

Correct option is C

Given:

​$$\frac{1}{\sec \theta - \tan \theta}$$​​

Formula Used:

Trigonometric identities:
​$$\sec \theta = \frac{1}{\cos \theta}\\\ \\\tan \theta = \frac{\sin \theta}{\cos \theta}$$​​

Solution:

​$$\frac{1}{\sec \theta - \tan \theta}\\\ \\= \frac{1}{\frac{1}{\cos \theta} - \frac{\sin \theta}{\cos \theta}}\\\ \\= \frac{1}{\frac{1 - \sin \theta}{\cos \theta}}\\\ \\= \frac{\cos \theta}{1 - \sin \theta}\\\ \\$$

$$= \frac{\cos \theta}{1 - \sin \theta}\times\frac{1 +\sin \theta}{1 +\sin \theta}\\\ \\$$

$$= \frac{(\cos \theta)(1 +\sin \theta)}{1 -\sin^2 \theta}\\\ \\$$

$$= \frac{(\cos \theta)(1 +\sin \theta)}{cos^2 \theta}\\\ \\$$

$$= \frac{(1 +\sin \theta)}{cos \theta}\\\ \\$$ 

$$= sec \theta +{tan \theta}\\\ \\$$​​

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