Trigonometric identity verification
Check the final answer first, then review the worked steps.
Problem
Trigonometric identity verification
Answer
\(\sin x + \cos x = \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)\)
Step-by-step solution
- Apply the sum formula for sine: The sum formula for sine is given by $\sin(A + B) = \sin A \cos B + \cos A \sin B$. We will apply this to the right side of the equation, $\sqrt{2} \sin \left(x + \frac{\pi}{4}\right)$. Here, $A = x$ and $B = \frac{\pi}{4}$.
- Substitute values into the sum formula: Substitute $A=x$ and $B=\frac{\pi}{4}$ into the formula: $\sqrt{2} \left( \sin x \cos \frac{\pi}{4} + \cos x \sin \frac{\pi}{4} \right)$.
- Evaluate trigonometric functions for $\frac{\pi}{4}$: We know that $\cos \frac{\pi}{4} = \frac{\sqrt{2}}{2}$ and $\sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$.
- Substitute the evaluated values: Substitute these values back into the expression from Step 2: $\sqrt{2} \left( \sin x \cdot \frac{\sqrt{2}}{2} + \cos x \cdot \frac{\sqrt{2}}{2} \right)$.
- Distribute $\sqrt{2}$: Distribute $\sqrt{2}$ to both terms inside the parentheses: $\sqrt{2} \cdot \sin x \cdot \frac{\sqrt{2}}{2} + \sqrt{2} \cdot \cos x \cdot \frac{\sqrt{2}}{2}$.
- Simplify the expression: Since $\sqrt{2} \cdot \sqrt{2} = 2$, the expression becomes: $\frac{2}{2} \sin x + \frac{2}{2} \cos x$, which simplifies to $\sin x + \cos x$.
- Conclusion: We have shown that the right side of the original equation, $\sqrt{2} \sin \left(x + \frac{\pi}{4}\right)$, simplifies to $\sin x + \cos x$, which is the left side of the equation. Therefore, the identity is verified.