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26. \( \frac{\cos ^{2}[(\pi / 2)-x]}{\cos x} \)

Ask by Salazar Carlson. in the United States
Feb 04,2025

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The simplified form is \( \sec(x) - \cos(x) \).

Solución

Calculate or simplify the expression \( \frac{\cos ^{2}[(\pi / 2)-x]}{\cos x} \). Simplify the expression by following steps: - step0: Solution: \(\frac{\cos^{2}\left(\frac{\pi }{2}-x\right)}{\cos\left(x\right)}\) - step1: Rewrite the expression: \(\frac{\sin^{2}\left(x\right)}{\cos\left(x\right)}\) - step2: Rewrite the expression: \(\sin^{2}\left(x\right)\cos^{-1}\left(x\right)\) - step3: Calculate: \(\left(1-\cos^{2}\left(x\right)\right)\cos^{-1}\left(x\right)\) - step4: Rewrite the expression: \(\cos^{-1}\left(x\right)-\cos\left(x\right)\) - step5: Simplify: \(\sec\left(x\right)-\cos\left(x\right)\) The simplified form of the expression \( \frac{\cos ^{2}[(\pi / 2)-x]}{\cos x} \) is \( \sec(x) - \cos(x) \).

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Bonus Knowledge

To simplify the expression \( \frac{\cos ^{2}[(\pi / 2)-x]}{\cos x} \), we can start by using the co-function identity. Based on this identity, we have: \[ \cos\left(\frac{\pi}{2} - x\right) = \sin x \] So, we rewrite the expression: \[ \cos^{2}\left(\frac{\pi}{2} - x\right) = \sin^{2}x \] Now, substituting this into our original expression: \[ \frac{\cos^{2}\left(\frac{\pi}{2} - x\right)}{\cos x} = \frac{\sin^{2} x}{\cos x} \] This can be rewritten using the identity for tangent: \[ \frac{\sin^{2} x}{\cos x} = \sin x \cdot \tan x \] Thus, the simplified form of the expression is: \[ \sin x \cdot \tan x \]

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