Answer:
See below:
Step-by-step explanation:
Let's use the trigonometric sum and difference formulas to verify the given expression:
[tex] \sin(30^\circ + \theta) + \cos(60^\circ + \theta) = \cos(\theta) [/tex]
We'll use the following trigonometric sum and difference formulas:
- [tex]\sin(A + B) = \sin A \cos B + \cos A \sin B[/tex]
- [tex]\cos(A + B) = \cos A \cos B - \sin A \sin B[/tex]
Now, substitute the given values into the expression:
Left hand side:
[tex] \sin(30^\circ + \theta) + \cos(60^\circ + \theta) [/tex]
Using the sum formula for sine and cosine.
[tex]= \left(\sin 30^\circ \cos \theta + \cos 30^\circ \sin \theta\right) + \left(\cos 60^\circ \cos \theta - \sin 60^\circ \sin \theta\right) [/tex]
Now, simplify each term:
[tex]= \left(\dfrac{1}{2} \cos \theta + \dfrac{\sqrt{3}}{2} \sin \theta\right) + \left(\dfrac{1}{2} \cos \theta - \dfrac{\sqrt{3}}{2} \sin \theta\right) [/tex]
Combine like terms:
[tex] = \dfrac{1}{2} \cos \theta + \dfrac{1}{2} \cos \theta [/tex]
[tex] = \cos \theta [/tex]
Right Hand side:
Therefore, the expression is equal to [tex]\cos \theta[/tex], verifying the given trigonometric identity.
