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Problem 63

Prove $\sin \left(45^{\circ}+\mathrm{x}\right)+\sin \left(45^{\circ}-\mathrm{x}\right)=\sqrt{2} \cos \mathrm{x}$

Expert verified
Using the sum-to-product formula for sine, we can expand and simplify the given expression as follows: 1. Apply the sum-to-product formula: $$\sin \left(45^{\circ}+\mathrm{x}\right) + \sin \left(45^{\circ}-\mathrm{x}\right) = (\sin(45^{\circ})\cos(\mathrm x) + \cos(45^{\circ})\sin(\mathrm x)) + (\sin(45^{\circ})\cos(\mathrm x) - \cos(45^{\circ})\sin(\mathrm x))$$ 2. Combine like terms: $$2\sin(45^{\circ})\cos(\mathrm x)$$ 3. Substitute the value of $$sin(45^\circ) = \frac{1}{\sqrt{2}}$$: $$\sqrt{2}\cos(\mathrm{x})$$ Hence, we've proved the identity $$\sin \left(45^{\circ}+\mathrm{x}\right)+\sin \left(45^{\circ}-\mathrm{x}\right)=\sqrt{2} \cos \mathrm{x}$$.
See the step by step solution

Step 1: Apply Sum-to-Product Formula for Sine

\ We will start by applying the sum-to-product formula for sine, which states: $\sin(A + B) = \sin A \cos B + \cos A \sin B$ Applying this formula to both terms of the given identity: $\sin \left(45^{\circ}+\mathrm{x}\right) = \sin(45^{\circ})\cos(\mathrm x) + \cos(45^{\circ})\sin(\mathrm x)$ and $\sin \left(45^{\circ}-\mathrm{x}\right) = \sin(45^{\circ})\cos(\mathrm x) - \cos(45^{\circ})\sin(\mathrm x)$

Step 2: Add the expanded expressions

\ Now, we will add the two expressions obtained in Step 1: $\sin \left(45^{\circ}+\mathrm{x}\right) + \sin \left(45^{\circ}-\mathrm{x}\right) = (\sin(45^{\circ})\cos(\mathrm x) + \cos(45^{\circ})\sin(\mathrm x)) + (\sin(45^{\circ})\cos(\mathrm x) - \cos(45^{\circ})\sin(\mathrm x))$ After combining like terms, we obtain: $\sin \left(45^{\circ}+\mathrm{x}\right) + \sin \left(45^{\circ}-\mathrm{x}\right) = 2\sin(45^{\circ})\cos(\mathrm x)$

Step 3: Simplify using the value of sin(45°)

\ We know that $$\sin(45^{\circ}) = \frac{1}{\sqrt{2}}$$. Substituting this value into the expression from Step 2: $2\sin(45^{\circ})\cos(\mathrm x) = 2 \cdot \frac{1}{\sqrt{2}} \cos(\mathrm x)$ Simplifying further, we get: $2\sin(45^{\circ})\cos(\mathrm x) = \sqrt{2}\cos(\mathrm x)$

Step 4: Concluding the proof

\ By following these steps, we have shown that $\sin \left(45^{\circ}+\mathrm{x}\right)+\sin \left(45^{\circ}-\mathrm{x}\right)=\sqrt{2} \cos \mathrm{x}$ which is the identity we were asked to prove. This completes the proof.

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