Prove the following identity-cos -cos -2-sin -sin -2-4cos2 - -2

To prove: (cos α+cos β)^2+(sin α+sin β)^2 =4cos^2 ( α β2 ) L.H.S. = nbsp;(cos α+cos β)^2+(sin α+sin β)^2 =cos^2 α+cos^2 β+2cos α cos nbsp; β+sin^2 α+sin^ ...

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Standard XI

Mathematics

Basic Trigonometric Identities

Prove the fol...

Question

Prove the following identity:

$(c o s α + c o s β)^{2} + (s i n α + s i n β)^{2}$
$= 4 c o s^{2} (\frac{α - β}{2})$

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Solution

To prove:
$(c o s α + c o s β)^{2} + (s i n α + s i n β)^{2}$
$= 4 c o s^{2} (\frac{α - β}{2})$

L.H.S. = $(c o s α + c o s β)^{2} + (s i n α + s i n β)^{2}$

$= c o s^{2} α + c o s^{2} β + 2 c o s α c o s β + s i n^{2} α + s i n^{2} β + 2 s i n α s i n β$

$= (c o s^{2} α + s i n^{2} α) + (c o s^{2} β + s i n^{2} β)$
$+ 2 (c o s α c o s β + s i n α s i n β)$

$= 1 + 1 + 2 c o s (α - β)$

$= 2 + 2 c o s (α - β)$

$= 2 {2 c o s^{2} (\frac{α - β}{2})} {∵ c o s 2 θ = 2 c o s^{2} θ - 1}$

$= 4 c o s^{2} (\frac{α - β}{2})$

= R.H.S.

Hence, proved

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${(c o s α + c o s β)}^{2} + {(s i n α + s i n β)}^{2} = 4 c o s^{2} (\frac{α - β}{2})$

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Basic Trigonometric Identities

Standard XI Mathematics

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Prove the following identity: (cos α+cosβ)2+(sinα+sinβ)2 =4cos2(α−β2)

Prove the following identity:

$(c o s α + c o s β)^{2} + (s i n α + s i n β)^{2}$
$= 4 c o s^{2} (\frac{α - β}{2})$