Question

Show that none of the following is an identity:
(i) cos²θ + cos θ = 1
(ii) sin2θ + sin θ = 2
(iii) tan²θ + sin θ = cos²θ

Answer 1

$\begin{array}{l}\\ \left(\text{i}\right){\text{cos}}^2\theta +\text{cos}\theta \text{=1}\\ {\text{LHS=cos}}^2\theta +\text{cos}\theta \\ \text{=1}-{\text{sin}}^2\theta +\text{cos}\theta \\ \text{=1}-({\text{sin}}^2\theta -\text{cos}\theta )\\ \text{Since LHS}\ne \text{RHS, this is not an identity.}\\ \\ \left(\text{ii}\right){\text{sin}}^2\theta +\text{sin}\theta \text{=1}\\ {\text{LHS=sin}}^2\theta +\text{sin}\theta \\ \text{=1}-{\text{cos}}^2\theta +\text{sin}\theta \\ \text{=1}-({\text{cos}}^2\theta -\text{sin}\theta )\\ \text{Since LHS≠RHS, this is not an identity}\text{.}\\ \\ \left(\text{iii}\right){\text{tan}}^2\theta +\text{sin}\theta {\text{=cos}}^2\theta \\ {\text{LHS=tan}}^2\theta +\text{sin}\theta \\ \text{=}\frac{{\text{sin}}^2\theta }{{\text{cos}}^2\theta }+\text{sin}\theta \\ \text{=}\frac{1-{\text{cos}}^2\theta }{{\text{cos}}^2\theta }+\text{sin}\theta \\ {\text{=sec}}^2\theta -1+\text{sin}\theta \\ \mathrm{Since}\mathrm{LHS}\ne \text{RHS, this is not an identity}\text{.}\\ \end{array}$