To simplify the expression cos(21°) tan²(2°), we start by breaking it into its components. Let’s first recall the definition of the tangent function:
- tan(θ) = sin(θ) / cos(θ)
Therefore, we can write:
- tan²(2°) = (sin(2°) / cos(2°))² = sin²(2°) / cos²(2°)
Now, substitute this into the original expression:
- cos(21°) tan²(2°) = cos(21°) * (sin²(2°) / cos²(2°))
This can be rewritten as:
- = (cos(21°) * sin²(2°)) / cos²(2°)
At this point, recognizing that we can’t simplify this expression significantly further without numerical values or specific trigonometric identities involving sine and cosine for these angles, we can note that cos(21°) is a specific value which is approximately 0.93358, and sin(2°) is approximately 0.0349, cos(2°) is approximately 0.99939.
Thus:
- cos(21°) tan²(2°) ≈ (0.93358 * (0.0349)²) / (0.99939)²
Calculating the approximate values gives:
- ≈ (0.93358 * 0.001219) / 0.99878 ≈ 0.0010964
So, the simplified form of cos(21°) tan²(2°) using numeric approximation gives you:
- ≈ 0.0010964
Remember, without further context or restrictions, this is a numerical approximation rather than a strict algebraic simplification. In summary, we explored the definition of tangent, transformed the expression, and then calculated an approximation based on known values.