If y+32y+5=sin2x+2cosx+1, then the value of y lies in the interval of
−∞,−83
−125,∞
−83,−125
−83,∞
We have
y+32y+5=sin2x+2cosx+1∴ cos2x−2cosx+1=3−y+32y+5∴ (cosx−1)2=5y+122y+5
now −1≤cosx≤1
∴ −2≤cosx−1≤0∴ 0≤(cosx−1)2≤4∴ 0≤5y+122y+5≤4∴ 0≤5y+122y+5 and 5y+122y+5≤4∴ 5y+122y+5≥0 and 3y+82y+5≥0∴ −∞,−52∪−125,∞ and −∞,−83∪−52,∞∴ y∈−∞,−83∪−125,∞