p.t $$ \cos 75^{\circ}= an 8^{\circ}=\frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} $$

Question

p.t $$ \cos 75^{\circ}=	an 8^{\circ}=\frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} $$

UpStudy AI · Accepted Answer

Given the equation $$ \cos 75^{\circ}=	an 8^{\circ}=\frac{\cos 7^{\circ}-\sin 7^{\circ}}{\cos 7^{\circ}+\sin 7^{\circ}} $$, 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:
  $$\frac{\left(\cos\left(7^{\circ}ight)-\sin\left(7^{\circ}ight)ight)}{\left(\cos\left(7^{\circ}ight)+\sin\left(7^{\circ}ight)ight)}$$
- step1: Remove the parentheses:
  $$\frac{\cos\left(7^{\circ}ight)-\sin\left(7^{\circ}ight)}{\cos\left(7^{\circ}ight)+\sin\left(7^{\circ}ight)}$$
- 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 $$ \cos 75^{\circ}=	an 8^{\circ}=0.781286 $$.

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 $$ \frac{\sqrt{6} - \sqrt{2}}{4} $$.
$$ \cos 75^{\circ} = \frac{\sqrt{6} - \sqrt{2}}{4} $$

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