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Trigonometric Identities and Equations\newlineSum and difference identities: Problem type 11: Degrees\newlineFind the exact value of \newlinecos⁡105∘\cos 105^\circ by using a sum or difference formula.

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Q. Trigonometric Identities and Equations\newlineSum and difference identities: Problem type 11: Degrees\newlineFind the exact value of \newlinecos⁡105∘\cos 105^\circ by using a sum or difference formula.
  1. Express as Sum of Angles: To find the exact value of cos⁡(105°)\cos(105°), we can express 105°105° as the sum or difference of angles for which we know the exact values of the cosine function. One way to do this is to express 105°105° as the sum of 60°60° and 45°45°, since the cosine values for these angles are known.
  2. Apply Sum Formula: Using the sum formula for cosine, which is cos⁡(A+B)=cos⁡(A)cos⁡(B)−sin⁡(A)sin⁡(B)\cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B), we can write cos⁡(105°)\cos(105°) as cos⁡(60°+45°)\cos(60° + 45°).
  3. Substitute Known Values: Now we substitute the known values of cos⁡(60°)\cos(60°), cos⁡(45°)\cos(45°), sin⁡(60°)\sin(60°), and sin⁡(45°)\sin(45°) into the formula. These values are:\newlinecos⁡(60°)=12\cos(60°) = \frac{1}{2},\newlinecos⁡(45°)=22\cos(45°) = \frac{\sqrt{2}}{2},\newlinesin⁡(60°)=32\sin(60°) = \frac{\sqrt{3}}{2},\newlinesin⁡(45°)=22\sin(45°) = \frac{\sqrt{2}}{2}.
  4. Calculate Using Formula: Plugging the values into the sum formula, we get:\newlinecos⁡(105°)=cos⁡(60°)cos⁡(45°)−sin⁡(60°)sin⁡(45°)\cos(105°) = \cos(60°)\cos(45°) - \sin(60°)\sin(45°)\newline =(12)(22)−(32)(22)= (\frac{1}{2})(\frac{\sqrt{2}}{2}) - (\frac{\sqrt{3}}{2})(\frac{\sqrt{2}}{2})\newline =24−64= \frac{\sqrt{2}}{4} - \frac{\sqrt{6}}{4}.
  5. Combine Terms: Combining the terms, we get: \newlinecos⁡(105°)=(2−6)/4\cos(105°) = (\sqrt{2} - \sqrt{6})/4.\newlineThis is the exact value of cos⁡(105°)\cos(105°) using the sum formula.

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