Simplifying and Understanding the Identity:

cos²x + 2cos x sec x + sec²x + sin²x + 2sin x csc x + csc²x

Are you struggling with a complex trigonometric expression like
cos²x + 2cos x sec x + sec²x + sin²x + 2sin x csc x + csc²x?
This article will walk you through simplifying and understanding this identity step-by-step, helping you master key trigonometric concepts efficiently.

Understanding the Context

Breaking Down the Expression

Let’s examine each term in the expression carefully:

cos²x + 2cos x sec x + sec²x

sin²x + 2sin x csc x + csc²x

We begin by recalling fundamental trigonometric identities:

$= \cos^2 x + 2\cos x \sec x + \sec^2 x + \sin^2 x + 2\sin x \csc x + \csc^2 x$

Key Insights

sec x = 1/cos x
csc x = 1/sin x

Using these definitions, let’s simplify each part.

Step 1: Simplify Terms Involving Secants and Cosecants

2cos x sec x = 2cos x × (1/cos x) = 2
2sin x csc x = 2sin x × (1/sin x) = 2

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Final Thoughts

So the expression reduces to:

cos²x + 2 + sec²x

sin²x + 2 + csc²x

Combine constants:

(cos²x + sin²x) + (sec²x + csc²x) + 4

Step 2: Apply Pythagorean Identity

We know from the Pythagorean identity:

cos²x + sin²x = 1

Now, rewrite the expression:

1 + sec²x + csc²x + 4 = sec²x + csc²x + 5