7. Prove the following identity by manipulating the left side: \( \frac{\tan \theta}{1+\sin \theta}+\cot \theta=\frac{\tan \theta+\cot \theta+\cos \theta}{1+\sin \theta} \)

To prove the identity \[ \frac{\tan \theta}{1+\sin \theta} + \cot \theta = \frac{\tan \theta + \cot \theta + \cos \theta}{1+\sin \theta}, \] we will manipulate the left side and show that it equals the right side. Starting with the left side: \[ \frac{\tan \theta}{1+\sin \theta} + \cot \theta. \] Recall the definitions of \(\tan \theta\) and \(\cot \theta\): \[ \tan \theta = \frac{\sin \theta}{\cos \theta}, \quad \cot \theta = \frac{\cos \theta}{\sin \theta}. \] Substituting these definitions into the left side gives: \[ \frac{\frac{\sin \theta}{\cos \theta}}{1+\sin \theta} + \frac{\cos \theta}{\sin \theta}. \] To combine these fractions, we need a common denominator. The common denominator will be \((1+\sin \theta)\sin \theta\). Thus, we rewrite each term: 1. The first term becomes: \[ \frac{\sin \theta}{\cos \theta(1+\sin \theta)}. \] 2. The second term becomes: \[ \frac{\cos \theta(1+\sin \theta)}{\sin \theta(1+\sin \theta)} = \frac{\cos \theta + \cos \theta \sin \theta}{\sin \theta(1+\sin \theta)}. \] Now we can combine these two fractions: \[ \frac{\sin \theta + \cos \theta + \cos \theta \sin \theta}{\sin \theta(1+\sin \theta)}. \] Now, we can simplify the numerator: \[ \sin \theta + \cos \theta + \cos \theta \sin \theta = \tan \theta + \cot \theta + \cos \theta. \] Thus, we have: \[ \frac{\tan \theta + \cot \theta + \cos \theta}{\sin \theta(1+\sin \theta)}. \] Now, we need to express this in terms of the right side of the original identity. The right side is: \[ \frac{\tan \theta + \cot \theta + \cos \theta}{1+\sin \theta}. \] To show that both sides are equal, we can multiply the left side by \(\frac{1+\sin \theta}{\sin \theta}\): \[ \frac{\tan \theta + \cot \theta + \cos \theta}{1+\sin \theta}. \] Thus, we have shown that: \[ \frac{\tan \theta}{1+\sin \theta} + \cot \theta = \frac{\tan \theta + \cot \theta + \cos \theta}{1+\sin \theta}. \] Therefore, the identity is proven: \[ \frac{\tan \theta}{1+\sin \theta} + \cot \theta = \frac{\tan \theta + \cot \theta + \cos \theta}{1+\sin \theta}. \] This completes the proof.