Step-by-step explanation:
[tex](sec \: x + sin \: x)cot \: x = csc \: x + cos \: x \\ \\ LHS = (sec \: x + sin \: x)cot \: x \\ = (sec \: x + sin \: x) \times \frac{cos \: x}{sin \: x} \\ \\ = sec \: x \times \frac{cos \: x}{sin \: x} + sin \: x \times \frac{cos \: x}{sin \: x} \\ \\ = \frac{1}{cos\: x } \times \frac{cos \: x}{sin \: x} + sin \: x \times \frac{cos \: x}{sin \: x} \\ \\ = \frac{1}{sin \: x} + cos \: x\\ \\ = csc \: x+ cos \: x \\ = RHS \\ \\ \therefore \: (sec \: x + sin \: x)cot \: x = csc \: x + cos \: x \\ thus \: proved[/tex]
