p.t \( \cos 75^{\circ}=\tan 8^{\circ}=\frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} \)

Given the equation \( \cos 75^{\circ}=\tan 8^{\circ}=\frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} \), we can start by simplifying the expression on the right-hand side of the equation. The expression \( \frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} \) can be simplified using trigonometric identities. Let's simplify this expression step by step. Calculate the value by following steps: - step0: Calculate: \(\frac{\left(\cos\left(7^{\circ}\right)-\sin\left(7^{\circ}\right)\right)}{\left(\cos\left(7^{\circ}\right)+\sin\left(7^{\circ}\right)\right)}\) - step1: Remove the parentheses: \(\frac{\cos\left(7^{\circ}\right)-\sin\left(7^{\circ}\right)}{\cos\left(7^{\circ}\right)+\sin\left(7^{\circ}\right)}\) - step2: Calculate the trigonometric value: \(0.781286\) The simplified expression is \( \frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} = 0.781286 \). Now, we have the equation \( \cos 75^{\circ}=\tan 8^{\circ}=0.781286 \). To find the value of \( \cos 75^{\circ} \), we can use the fact that \( \cos 75^{\circ} = \cos (45^{\circ} + 30^{\circ}) \). Using the angle addition formula for cosine, we have \( \cos (45^{\circ} + 30^{\circ}) = \cos 45^{\circ} \cos 30^{\circ} - \sin 45^{\circ} \sin 30^{\circ} \). Substituting the known values, we get \( \cos 75^{\circ} = \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2} \). Simplifying this expression, we find \( \cos 75^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4} \). Therefore, the value of \( \cos 75^{\circ} \) is \( \frac{\sqrt{6} - \sqrt{2}}{4} \).