= \cos^2 x + 2\cos x \sec x + \sec^2 x + \sin^2 x + 2\sin x \csc x + \csc^2 x - Nurtured Nest
Simplifying and Understanding the Identity:
cos²x + 2cos x sec x + sec²x + sin²x + 2sin x csc x + csc²x
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:
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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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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
