The set of values of  λ∈R such that sin2⁡θ+cos⁡θ=λcos2⁡θ holds for some θ is

We have tan2⁡θ+sec⁡θ=λ⇒sec2⁡θ+sec⁡θ−(λ+1)=0⇒ sec⁡θ=−1±4λ+52For real  secθ4λ+5≥0⇒λ≥−54-----1 Also, sec⁡θ≥1 or sec⁡θ≤−1⇒ −1±4λ+52≥1 or −1±4λ+52≤−1⇒ 4λ+5≥9 or 4λ+5≥1⇒ λ≥1 or λ≥−1----2∴  From (1) and (2), we get λ∈[−1,∞).