Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\sin \left(\frac{\pi}{2}+x\right)=\cos x$$

Step-by-Step Solution

Answer

The left-hand side of the equation \(\sin \left(\frac{\pi}{2}+x\right)\) does indeed simplify to \(\cos x\). Therefore, \(\sin \left(\frac{\pi}{2}+x\right) = \cos x\) is a true identity.

Step 1: Understand and use the Addition Formula

The addition formula for sine states that \(\sin(a+b) = \sin a \cos b + \cos a \sin b\). Therefore, we can write \(\sin \left(\frac{\pi}{2}+x\right)\) as \(\sin(\frac{\pi}{2})\cos x + \cos(\frac{\pi}{2})\sin x\).

Step 2: Substitute the values

Substitute the values of \(\sin(\frac{\pi}{2})\) and \(\cos(\frac{\pi}{2})\), which are 1 and 0 respectively. Therefore, the expression becomes \(1 * \cos x + 0 * \sin x\). This simplifies to \(\cos x\).

Step 3: Endpoint Verification

The right-hand side of our initial identity was \(\cos x\). Now the left-hand side, \(\sin \left(\frac{\pi}{2}+x\right)\), has also been simplified down to \(\cos x\) showing that the two sides are indeed equal and that verifies the identity.