Show that none of the following is an identity:

Solution:

(i) $\cos ^{2} \theta+\cos \theta=1$

$\mathrm{LHS}=\cos ^{2} \theta+\cos \theta$

$=1-\sin ^{2} \theta+\cos \theta$

$=1-\left(\sin ^{2} \theta-\cos \theta\right)$

Since $\mathrm{LHS} \neq \mathrm{RHS}$, this is not an identity.

(ii) $\sin ^{2} \theta+\sin \theta=1$

$\mathrm{LHS}=\sin ^{2} \theta+\sin \theta$

$=1-\cos ^{2} \theta+\sin \theta$+

$=1-\left(\cos ^{2} \theta-\sin \theta\right)$

Since LHS $\neq$ RHS, this is not an identity.

(iii) $\tan ^{2} \theta+\sin \theta=\cos ^{2} \theta$

$\mathrm{LHS}=\tan ^{2} \theta+\sin \theta$

$=\frac{\sin ^{2} \theta}{\cos ^{2} \theta}+\sin \theta$

$=\frac{1-\cos ^{2} \theta}{\cos ^{2} \theta}+\sin \theta$

$=\sec ^{2} \theta-1+\sin \theta$

Since LHS $\neq$ RHS, this is not an identity.