If cos A = 1/2 (a +1/a), show that: cos2A = 1/2 (a² +1/a²) | SciPiPupil

Question: If $cos A$ = $\dfrac{1}{2} \left ( a +\dfrac{1}{a} \right)$ , show that:

$cos 2A = \dfrac{1}{2} \left ( a² + \dfrac{1}{a²} \right)$

Solution:

Given,

$cos A$ =

$\dfrac{1}{2} \left ( a +\dfrac{1}{a} \right)$

We know,

$cos 2A$

$= 2cos²A - 1$

$= 2 (cosA)² -1$

$= 2 \left \{ \dfrac{1}{2} \left ( a + \dfrac{1}{a} \right) \right \}^2 -1$

$= 2 \left \{ \dfrac{1}{4} \left ( a² + 2 + \dfrac{1}{a²} \right) \right \} -1$

$= \dfrac{ \left ( a² + 2 + \dfrac{1}{a²} \right) }{2} -1$

$= \dfrac{ a² +2 +\dfrac{1}{a²} -2 }{2}$

$= \dfrac{a² +\dfrac{1}{a²}$

$= \dfrac{1}{2} \left ( a² + \dfrac{1}{a²} \right )$

= RHS

If cos A = 1/2 (a +1/a), show that: cos2A = 1/2 (a² +1/a²) | SciPiPupil