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Step-by-step explanation:
Let's simplify the expression:
(tan2x + 1)(cos²x - 1)
= (tan2x + 1)(cos²x - 1) (using the distributive property)
= tan2x * cos²x - tan2x * 1 + 1 * cos²x - 1 * 1
= tan2x * cos²x - tan2x + cos²x - 1
Now, let's simplify the terms:
tan2x * cos²x = tan2x * (1 - sin²x) (using the trigonometric identity)
= tan2x - tan2x * sin²x
= tan2x - (sin²x * tan2x)
= tan2x - sin²x * tan2x
Now, we can combine like terms:
tan2x - sin²x * tan2x = tan2x(1 - sin²x)
= tan2x * cos²x
So, the simplified expression is:
tan2x * cos²x - tan2x + cos²x - 1
= (tan2x * cos²x) - (tan2x) + (cos²x) - 1
= (tan2x * cos²x) - (tan2x) + (cos²x) - 1