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If tan(x)=(5)/(6) (in Quadrant-I), find cos(2x)=

If tan⁡(x)=56 \tan (x)=\frac{5}{6} (in Quadrant-I), find \newlinecos⁡(2x)= \cos (2 x)=

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Full solution

Q. If tan⁡(x)=56 \tan (x)=\frac{5}{6} (in Quadrant-I), find \newlinecos⁡(2x)= \cos (2 x)=
  1. Use cos⁡(2x)\cos(2x) identity: Use the identity for cos⁡(2x)\cos(2x) which is cos⁡(2x)=1−2sin⁡2(x)\cos(2x) = 1 - 2\sin^2(x). First, find sin⁡(x)\sin(x) using tan⁡(x)=sin⁡(x)cos⁡(x)\tan(x) = \frac{\sin(x)}{\cos(x)}.
  2. Calculate sin⁡(x)\sin(x) and \cos(x): Calculate \(\sin(x) and cos⁡(x)\cos(x) using k=161k = \frac{1}{\sqrt{61}}.
  3. Substitute sin⁡(x)\sin(x) into identity: Substitute sin⁡(x)\sin(x) into the cos⁡(2x)\cos(2x) identity.

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