Question
2) \( \tan ^{2} x\left(1-\sin ^{2} x\right) \)
Ask by Sanders Delgado. in South Africa
Feb 24,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( \sin^2 x \).
Solution
We start with the expression:
tan²x (1 – sin²x)
Recall that tan x = sin x/cos x, so tan²x = sin²x/cos²x. Substitute this into the expression:
= (sin²x/cos²x) (1 – sin²x)
Recognize that 1 – sin²x is equal to cos²x (by the Pythagorean identity sin²x + cos²x = 1). So we get:
= (sin²x/cos²x) (cos²x)
The cos²x cancels:
= sin²x
Thus, the simplified form of the expression is:
sin²x
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Simplify this solution Extra Insights
This expression can be further simplified using trigonometric identities. Specifically, we know that \( 1 - \sin^2 x = \cos^2 x \). Therefore, \( \tan^2 x(1 - \sin^2 x) \) can be rewritten as \( \tan^2 x \cdot \cos^2 x = \frac{\sin^2 x}{\cos^2 x} \cdot \cos^2 x = \sin^2 x \). So, the expression simplifies neatly to \( \sin^2 x \)! An interesting fact about the tangent function is that it's periodic with a period of \( \pi \). This means that for any angle \( x \), \( \tan(x + n\pi) = \tan x \) where \( n \) is any integer. This property is particularly useful when solving trigonometric equations or analyzing graphs. Keep this in mind as you delve into the world of tangents!
