Problem 78 - Algebra and Trigonometry Real Mathematics, Real People

Question

Verify the identity. $$\tan \left(\cos ^{-1} \frac{x+1}{2}\right)=\frac{\sqrt{4-(x+1)^{2}}}{x+1}$$

Step-by-Step Solution

Answer

\(\tan \left(\cos ^{-1} \frac{x+1}{2}\right)=\frac{\sqrt{4-(x+1)^{2}}}{x+1}\) is a valid identity.

Step 1: Express \(\cos ^{-1} \)

Instead of taking the cosine of the fraction \( \frac{x+1}{2} \), we'll interpret this as the angle whose cosine is \( \frac{x+1}{2} \). We'll call this angle \( \theta \), that is, \( \theta = \cos ^{-1} \frac{x+1}{2} \). Therefore, the problem can be rewritten as \( \tan( \theta ) = \frac{\sqrt{4-(x+1)^{2}}}{x+1} \).

Step 2: Use the relationship between cosine and sine

Since we know the cosine of \( \theta \), by using the Pythagorean identity \( \sin ^2(\theta) = 1 - \cos^2(\theta) \), we can compute that \( \sin(\theta) \) is \( \sqrt{1 - \cos ^2 (\theta)} = \sqrt{1 - (\frac{x+1}{2})^2} \) . This will get us closer to expressing the left side of the equation as a tan function.

Step 3: Convert cosine to tangent

Since \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), we can substitute the values of \( \sin(\theta) \) and \( \cos(\theta) \) that we found in the previous steps into this equation, yielding: \( \tan(\theta) = \frac{\sqrt{1 - (\frac{x+1}{2})^2}}{\frac{x+1}{2}} \) . This simplifies to \( \tan(\theta) = \frac{2\sqrt{1 - (\frac{x+1}{2})^2}}{x+1} \).

Step 4: Simplify the equation

To simplify the above expression further, we can multiply numerator and denominator by 2. This gives us: \( \tan(\theta) = \frac{\sqrt{4 - (x+1)^2}}{x+1} \). This matches the right-hand side of the original problem, verifying the identity.