The exercises in the image contain two problems from complex numbers and polynomial factorization.

Exercise 7.34:

Problem: Solve in $\mathbb{C}$ the equation: $(z + 1 + i)^n = 1$ This problem involves finding complex solutions $z$ such that when $z + 1 + i$ is raised to the power $n$, it equals 1. Typically, this could involve representing $z + 1 + i$ in polar form and solving for $z$ based on $n$-th roots of unity.

Exercise 7.35:

Problem: Given $n \in \mathbb{N}^*$, and $P_n = (X + 1)^n - (X - 1)^n$:

1. Factorize $P_n$ in $\mathbb{C}[X]$.

2. Deduce that: $\forall p \in \mathbb{N}^*, \prod_{k=1}^p \tan \frac{k \pi}{2p + 1} = \sqrt{2p + 1}$

   • Part 1: Factorizing $P_n$ requires using binomial expansions and possibly simplifying the expression by grouping terms.
   • Part 2: This is a trigonometric product identity, where the result is connected to properties of tangent functions and their roots over certain intervals.

Would you like a detailed solution to each part, or would you prefer explanations on specific concepts involved?

Related Questions:

1. What are the $n$-th roots of unity and how are they calculated?
2. How can complex numbers be represented in polar form for easier manipulation in powers?
3. What are the properties of binomial expansions and how do they aid in factorizing polynomials?
4. How can trigonometric identities be used to simplify products involving tangent functions?
5. What techniques exist for deducing product formulas in trigonometry?

Tip: When solving complex equations involving powers, converting to polar form can simplify calculations significantly.