Determine the exact value of cos(pi/8) and tan(pi/4 + pi/4) using the double angle formula.

Steps to Solve

1. Identify the Given Values Given that: $$ \cos A = \frac{\sqrt{2}}{2} $$ We also know that: $$ \tan A = \frac{\sin A}{\cos A} $$

2. Use the Double Angle Formulas The double angle formulas for cosine and tangent are:

• Cosine: $$ \cos(2A) = 2 \cos^2(A) - 1 $$
• Tangent: $$ \tan(2A) = \frac{2 \tan(A)}{1 - \tan^2(A)} $$

3. Calculate Cosine of Double Angle First, calculate $\cos(2A)$: Substituting $\cos A = \frac{\sqrt{2}}{2}$ into the formula: $$ \cos(2A) = 2 \left(\frac{\sqrt{2}}{2}\right)^2 - 1 $$ $$ = 2 \cdot \frac{2}{4} - 1 = 1 - 1 = 0 $$

4. Calculate Tangent of Double Angle Now, calculate $\tan(2A)$: To find $\tan A$, use $\sin^2 A + \cos^2 A = 1$: Since $\cos A = \frac{\sqrt{2}}{2}$, we find $\sin A$: $$ \sin A = \sqrt{1 - \left(\frac{\sqrt{2}}{2}\right)^2} = \sqrt{1 - \frac{2}{4}} = \sqrt{\frac{2}{4}} = \frac{\sqrt{2}}{2} $$

Thus, $$ \tan A = \frac{\sin A}{\cos A} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1 $$

Now substitute into $\tan(2A)$: $$ \tan(2A) = \frac{2(1)}{1 - (1)^2} = \frac{2}{0} $$

This indicates that $\tan(2A)$ is undefined (or approaches infinity).