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.

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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:

• sec x = 1/cos x
• csc x = 1/sin x

Using these definitions, let’s simplify each part.

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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

So the expression reduces to:

cos²x + 2 + sec²x

• sin²x + 2 + csc²x

Combine constants:

(cos²x + sin²x) + (sec²x + csc²x) + 4

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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

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Step 3: Express sec²x and csc²x Using Fundamental Identities

Using identities:

• sec²x = 1 + tan²x
• csc²x = 1 + cot²x

Substitute:

(1 + tan²x) + (1 + cot²x) + 5 = tan²x + cot²x + 7

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Alternative Simplified Form

Putting it all together, the original expression simplifies exactly to:

cos²x + 2cos x sec x + sec²x + sin²x + 2sin x csc x + csc²x =
tan²x + cot²x + 7

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Why This Identity Matters

Understanding such identities strengthens your foundation in trigonometry, especially useful in calculus, physics, and engineering when analyzing wave functions, oscillations, and circular motion.

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Summary

• Replace sec x and csc x with their reciprocal forms.
• Use the identity cos²x + sin²x = 1.
• Simplify cross terms using sec x = 1/cos x and csc x = 1/sin x.
• Apply Pythagorean identities to reduce complexity.
• Final simplified form: tan²x + cot²x + 7

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Final Thoughts

Trigonometric identities often hide elegant structures beneath complex-looking expressions. With systematic simplification — via reciprocals, identities, and algebra — even dense forms become clear. Mastery of identities like this empowers you to confidently tackle advanced calculus and problem-solving challenges.

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Try practicing: Simplify: cos²x + 2cos x sec x + sec²x + sin²x + 2sin x csc x + csc²x Using the steps above — you’ll quickly reach tan²x + cot²x + 7!

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