https://file.aiquickdraw.com/mathbot/file/1726974531681gfxu2h05.jpg

The given equation is:

\[
(1 + \tan b)^2 + (1 + \cot b)^2 = (\sec b + \csc b)^2
\]

Let's verify if this identity holds true by simplifying both sides.

### Left-hand side (LHS):
\[
(1 + \tan b)^2 + (1 + \cot b)^2
\]

Expanding both terms individually:

1. \((1 + \tan b)^2 = 1^2 + 2(1)(\tan b) + \tan^2 b = 1 + 2\tan b + \tan^2 b\)
2. \((1 + \cot b)^2 = 1^2 + 2(1)(\cot b) + \cot^2 b = 1 + 2\cot b + \cot^2 b\)

Thus, the LHS becomes:
\[
(1 + 2\tan b + \tan^2 b) + (1 + 2\cot b + \cot^2 b) = 2 + 2\tan b + \tan^2 b + 2\cot b + \cot^2 b
\]

### Right-hand side (RHS):
\[
(\sec b + \csc b)^2
\]

Expanding this:

\[
(\sec b + \csc b)^2 = \sec^2 b + 2\sec b \csc b + \csc^2 b
\]

Now, we need to show that both sides are equal.

Let me know if you'd like a more detailed step-by-step verification for this identity, or if you have any specific questions on this.

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Here are 5 related questions you might consider:

1. What is the relationship between \(\tan b\), \(\sec b\), and \(\cot b\)?
2. How can trigonometric identities help simplify complex expressions?
3. What are some key identities involving \(\sec b\) and \(\csc b\)?
4. How does expanding algebraic expressions relate to verifying trigonometric identities?
5. What role do Pythagorean identities play in simplifying trigonometric equations?

**Tip**: Always try to recognize patterns in trigonometric identities. This helps in quickly simplifying expressions!

(1 + tan b)^2 + (1 + cot b)^2 = (sec b + csc b)^2

Learn how to verify the trigonometric identity (1 + tan b)^2 + (1 + cot b)^2 = (sec b + csc b)^2 by expanding and simplifying both sides. This problem involves key trigonometric identities and algebraic expressions, suitable for Grades 11-12 students.

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