\frac{1-\cos \alpha}{1+\cos \alpha}=\tan ^2 \frac{\alpha}{2}
Answer (2)
Express tan 2 2 α as c o s 2 2 α s i n 2 2 α .
Apply the half-angle formulas: sin 2 2 α = 2 1 − c o s α and cos 2 2 α = 2 1 + c o s α .
Substitute the half-angle formulas into the expression.
Simplify the expression to obtain 1 + c o s α 1 − c o s α .
Conclude that 1 + c o s α 1 − c o s α = tan 2 2 α .
Explanation
Understanding the Problem We are given the trigonometric identity 1 + c o s α 1 − c o s α = tan 2 2 α and we want to prove that it is true.
Expressing tan in terms of sin and cos We will start from the right-hand side of the equation and use the half-angle formulas to express tan 2 2 α in terms of cos α . Recall that tan x = c o s x s i n x , so we have tan 2 2 α = cos 2 2 α sin 2 2 α
Applying Half-Angle Formulas Now, we use the half-angle formulas for sine and cosine, which are: sin 2 2 α = 2 1 − cos α cos 2 2 α = 2 1 + cos α
Substitution Substitute these half-angle formulas into the expression for tan 2 2 α :
tan 2 2 α = 2 1 + c o s α 2 1 − c o s α
Simplification Simplify the expression by canceling the factor of 2 in the numerator and denominator: tan 2 2 α = 1 + cos α 1 − cos α
Conclusion Thus, we have shown that 1 + c o s α 1 − c o s α = tan 2 2 α .
Examples
Trigonometric identities are useful in various fields such as physics, engineering, and computer graphics. For example, in physics, they can be used to simplify expressions involving angles in wave mechanics or optics. In computer graphics, they are used to perform rotations and transformations of objects in 3D space. Knowing how to manipulate and prove trigonometric identities allows us to solve problems more efficiently in these fields.
The identity 1 + c o s α 1 − c o s α = tan 2 2 α can be proved using half-angle formulas for sine and cosine. By substituting these formulas into the tangent expression and simplifying, we confirm the identity to be true. This understanding is crucial in trigonometry and has applications in various scientific fields.
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