Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\tan \left(\frac{\pi}{4}-\theta\right)=\frac{1-\tan \theta}{1+\tan \theta}$$

Step-by-Step Solution

Answer

The given identity \(\tan \left(\frac{\pi}{4}-\theta\right)=\frac{1-\tan \theta}{1+\tan \theta}\) has been verified.

Step 1: Use Trigonometric Identity

Recall that for any real number \(x\), \(\tan(x)\) is defined as the quotient of the sine and cosine of \(x\). Therefore, express \(\tan \left(\frac{\pi}{4}-\theta\right)\) as \(\frac{\sin \left(\frac{\pi}{4}-\theta\right)}{\cos \left(\frac{\pi}{4}-\theta\right)}\).

Step 2: Apply the Sine and Cosine Difference Identities

The sine and cosine difference identities are \(\sin(a-b)=\sin(a)\cos(b) - \cos(a)\sin(b)\) and \(\cos(a-b)=\cos(a)\cos(b) + \sin(a)\sin(b)\). Apply these identities to both the numerator and the denominator of the previous expression with \(a=\frac{\pi}{4}\) and \(b=\theta\). This gives \(\frac{\sin \left(\frac{\pi}{4}\right)\cos \left(\theta\right) - \cos \left(\frac{\pi}{4}\right)\sin \left(\theta\right)}{\cos \left(\frac{\pi}{4}\right)\cos \left(\theta\right) + \sin \left(\frac{\pi}{4}\right)\sin \left(\theta\right)}\).

Step 3: Evaluate the Known Quantities

For any real number \(x\), \(\sin \left(\frac{\pi}{4}\right) = \cos \left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\). Substitute these values into the previous expression and simplify to obtain \(\frac{1 - \tan(\theta)}{1 + \tan(\theta)}\)

Step 4: Proof Of The Identity

Since the right side of the identity was obtained from the left side through basic algebraic manipulations and the use of known trigonometric identities, hence the identity \(\tan \left(\frac{\pi}{4}-\theta\right)=\frac{1-\tan \theta}{1+\tan \theta}\) has been confirmed.