Show that: [tex]$2 \sin ^2\left(\frac{\pi}{4}-\frac{\theta}{2}\right)=1-\sin \theta$[/tex]
Answer (1)
Use the identity sin 2 ( x ) = 2 1 − c o s ( 2 x ) to rewrite the left-hand side.
Simplify the expression to 1 − cos ( 2 π − θ ) .
Apply the identity cos ( 2 π − x ) = sin ( x ) to get 1 − sin ( θ ) .
The simplified LHS matches the RHS, thus 2 sin 2 ( 4 π − 2 θ ) = 1 − sin θ is proven. 2 sin 2 ( 4 π − 2 θ ) = 1 − sin θ
Explanation
Understanding the Problem We are tasked with proving the trigonometric identity 2 sin 2 ( 4 π − 2 θ ) = 1 − sin θ . This involves manipulating the left-hand side (LHS) of the equation to show that it is equal to the right-hand side (RHS). We will use trigonometric identities to simplify the LHS until it matches the RHS.
Applying the Sine Squared Identity We'll start by using the identity sin 2 ( x ) = 2 1 − c o s ( 2 x ) to rewrite the left-hand side (LHS) of the equation.
Substituting into the Identity Applying the identity to the LHS, we get:
2 sin 2 ( 4 π − 2 θ ) = 2 ⋅ 2 1 − c o s ( 2 ( 4 π − 2 θ ) )
Simplifying the Cosine Argument Simplifying the expression, we have:
2 ⋅ 2 1 − c o s ( 2 ( 4 π − 2 θ ) ) = 1 − cos ( 2 π − θ )
Using the Complementary Angle Identity Now, we use the identity cos ( 2 π − x ) = sin ( x ) to further simplify the expression.
Simplifying with the Complementary Angle Identity Applying the identity, we get:
1 − cos ( 2 π − θ ) = 1 − sin ( θ )
Conclusion Comparing the simplified LHS with the right-hand side (RHS) of the original equation, we see that they are equal. Therefore, the identity is verified.
2 sin 2 ( 4 π − 2 θ ) = 1 − sin θ
Examples
Trigonometric identities are useful in physics, especially when dealing with wave phenomena. For example, when analyzing the interference patterns of light waves, identities like the one we proved can help simplify complex expressions and make calculations easier. Imagine designing an optical instrument where you need to predict how light will behave; these identities become essential tools in your problem-solving toolkit. They allow you to manipulate equations and find the relationships between different parameters of the wave, such as amplitude and phase.
