A Prove that Sin2 O + Cos2 O = Cos2 O + Sin2 O b Prove that (1 + Cos O) / Sin O = Cosec O + Cot O

The fundamental trigonometric identity states that for any angle $\theta$:

$$\sin^2 \theta + \cos^2 \theta = 1$$

The order of addition does not change the sum. Therefore, $\cos^2 \theta + \sin^2 \theta$ is also equal to 1.

$$\cos^2 \theta + \sin^2 \theta = 1$$

Since both sides of the equation are equal to 1, they are equal to each other.

$$\sin^2 \theta + \cos^2 \theta = \cos^2 \theta + \sin^2 \theta$$

This statement is an identity based on the commutative property of addition.

$$\boxed{\text{Proven}}$$

Recall that $\csc \theta = \frac{1}{\sin \theta}$ and $\cot \theta = \frac{\cos \theta}{\sin \theta}$.

$$\text{LHS} = \csc \theta + \cot \theta$$

The LHS now matches the RHS.

$$\csc \theta + \cot \theta = \text{RHS}$$ $$\boxed{\text{Proven}}$$