If secα is  the average of sec⁡(α−2β) and sec⁡(α+2β) then the value of 2sin2⁡β−sin2⁡α is

sec⁡α=sec⁡(α−2β)+sec⁡(α+2β)2⇒ 2cos⁡α=cos⁡(α+2β)+cos⁡(α−2β)cos⁡(α−2β)cos⁡(α+2β)⇒ cos⁡2α+cos⁡4β=cos⁡α(2cos⁡α⋅cos⁡2β)⇒ 2cos2⁡α−1+2cos2⁡2β−1=2cos2⁡αcos⁡2β⇒ cos2⁡α(1−cos⁡2β)+(cos⁡2β+1)(cos⁡2β−1)=0⇒ (1−cos⁡2β)cos2⁡α−cos⁡2β−1=0⇒ cos2⁡α=cos⁡2β+1 ( as β≠nπ)⇒ cos2⁡α=2cos2⁡β⇒ 1−sin2⁡α=21−sin2⁡β⇒ 2sin2⁡β−sin2⁡α=1