Question

prove that 1 + cosA= 2cos^2A/2
​

Answer 1

SOLUTION

TO PROVE

$\displaystyle \sf{1 + \cos A = 2 { \cos}^{2} \: \frac{A}{2} }$

EVALUATION

We are aware of the Trigonometric formula that

$\displaystyle \sf{ \cos( \alpha + \beta ) = \cos \alpha \cos \beta - \sin \alpha \sin \beta }$

$\displaystyle \sf{ Putting \: \: \alpha = \beta = \frac{A}{2} }$

We get

$\displaystyle \sf{ \cos \bigg(\frac{A}{2} + \frac{A}{2} \bigg) = \cos \frac{A}{2} \cos \frac{A}{2} - \sin \frac{A}{2}\sin \frac{A}{2}}$

$\displaystyle \sf{ \implies \: \cos A = {\cos }^{2} \frac{A}{2} -{ \sin}^{2} \frac{A}{2}}$

$\displaystyle \sf{ \implies \: \cos A = {\cos }^{2} \frac{A}{2} - \bigg(1 - { \cos}^{2} \frac{A}{2} \bigg)}$

$\displaystyle \sf{ \implies \: \cos A = {\cos }^{2} \frac{A}{2} - 1 + { \cos}^{2} \frac{A}{2} }$

$\displaystyle \sf{ \implies \: \cos A = 2{\cos }^{2} \frac{A}{2} - 1 }$

$\displaystyle \sf{ \implies \: 1 + \cos A = 2{\cos }^{2} \frac{A}{2} }$

Hence proved

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