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Proving the Trigonometric Identity: (tan²x + 1)(cos²x + 1) = tan²x + 2
This article will demonstrate the steps involved in proving the trigonometric identity: (tan²x + 1)(cos²x + 1) = tan²x + 2.
Utilizing Fundamental Identities
We begin by expanding the left-hand side of the equation using the distributive property:
(tan²x + 1)(cos²x + 1) = tan²x * cos²x + tan²x + cos²x + 1
Next, we employ the fundamental trigonometric identity tan²x + 1 = sec²x:
= sec²x * cos²x + tan²x + cos²x + 1
Recall that secx = 1/cosx, so sec²x * cos²x = 1:
= 1 + tan²x + cos²x + 1
Finally, we utilize another fundamental identity, sin²x + cos²x = 1, to simplify further:
= 1 + tan²x + (1 - sin²x) + 1
Combining like terms, we arrive at:
= tan²x + 2
Conclusion
We have successfully shown that the left-hand side of the equation simplifies to tan²x + 2, which is the right-hand side of the equation. Therefore, the trigonometric identity (tan²x + 1)(cos²x + 1) = tan²x + 2 is proven.