Simplify (1 – tan²x)tan 2x tan x
The Correct Answer and Explanation is :
To simplify the given expression:
[
(1 – \tan^2x)\tan 2x \tan x
]
Step 1: Use the Double-Angle Identity for (\tan 2x)
The double-angle identity for tangent is:
[
\tan 2x = \frac{2 \tan x}{1 – \tan^2x}
]
Substitute this into the expression:
[
(1 – \tan^2x)\cdot \frac{2 \tan x}{1 – \tan^2x} \cdot \tan x
]
Step 2: Simplify the expression
The factor (1 – \tan^2x) cancels out from the numerator and denominator:
[
\frac{(1 – \tan^2x)}{(1 – \tan^2x)} = 1
]
This leaves:
[
2 \tan x \cdot \tan x
]
Step 3: Simplify further
[
2 \tan x \cdot \tan x = 2 \tan^2x
]
Thus, the simplified expression is:
[
2 \tan^2x
]
Explanation
The key to simplifying this expression is recognizing and using the trigonometric identity for (\tan 2x), which relates the double angle of tangent to its single angle counterpart. By substituting (\tan 2x = \frac{2 \tan x}{1 – \tan^2x}), the factor (1 – \tan^2x) cancels out with the denominator.
This simplification reduces the complexity of the expression, showing how the original structure condenses to a simpler form: (2 \tan^2x). The steps demonstrate how trigonometric identities can transform and simplify expressions, which is a common technique in solving more complex trigonometric equations or simplifying expressions in calculus and algebra.