Question

1. prove the following identities.

a.) \\( \sec \theta = \tan \theta + \frac{\cos \theta}{1 + \sin \theta} \\)
b.) \\( \frac{\sin \theta - \sin^3 \theta + \cos \theta}{\sin \theta} = \cos^2 \theta + \cot \theta \\)

Explanation:

Part a:

Step1: Rewrite right-hand side (RHS)

$RHS = \tan\theta + \frac{\cos\theta}{1+\sin\theta} = \frac{\sin\theta}{\cos\theta} + \frac{\cos\theta}{1+\sin\theta}$

Step2: Combine the fractions

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

Step3: Expand numerator

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

Step4: Use $\sin^2\theta+\cos^2\theta=1$

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

Step5: Simplify fraction

$=\frac{1}{\cos\theta} = \sec\theta = LHS$

Part b:

Step1: Split left-hand side (LHS)

$LHS = \frac{\sin\theta - \sin^3\theta + \cos\theta}{\sin\theta} = \frac{\sin\theta(1-\sin^2\theta) + \cos\theta}{\sin\theta}$

Step2: Use $\cos^2\theta=1-\sin^2\theta$

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

Step3: Split the fraction

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

Step4: Simplify each term

$=\cos^2\theta + \cot\theta = RHS$

Answer:

a.) The identity is proven.
b.) The identity is proven.