Show that none of the following is an identity:(i) cos2θ + cosθ = 1�

1.	Show that none of the following is an identity:(i) cos2θ + cosθ = 1 (ii) sin2θ + sinθ = 1(iii) tan2θ + sinθ = cos2θ
Answer» (i) cos²θ + cosθ = 1 LHS = cos² θ + cos θ =1 − sin² θ + cos θ = 1 − (sin² θ − cos θ) Since LHS ≠ RHS, this not an identity. (ii) sin²θ + sinθ = 1 LHS = sin² θ + sin θ = 1 − cos² θ + sin θ = 1 − (cos²θ − sin θ) Since LHS ≠ RHS, this is not an identity. (iii) tan²θ + sinθ = cos²θ LHS = tan² θ + sin θ = \(\frac{sin^2θ}{cos^2θ}\) + sinθ = \(\frac{1-cos^2θ}{cos^2θ}\)+ sinθ = sec² θ − 1 + sin θ Since LHS ≠ RHS, this is not an identity.

Show that none of the following is an identity:(i) cos2θ + cosθ = 1 (ii) sin2θ + sinθ = 1(iii) tan2θ + sinθ = cos2θ

Answer»

(i) cos²θ + cosθ = 1

LHS = cos² θ + cos θ

=1 − sin² θ + cos θ

= 1 − (sin² θ − cos θ)

Since LHS ≠ RHS, this not an identity.

(ii) sin²θ + sinθ = 1

LHS = sin² θ + sin θ

= 1 − cos² θ + sin θ

= 1 − (cos²θ − sin θ)

Since LHS ≠ RHS, this is not an identity.

(iii) tan²θ + sinθ = cos²θ

LHS = tan² θ + sin θ

= \(\frac{sin^2θ}{cos^2θ}\) + sinθ

= \(\frac{1-cos^2θ}{cos^2θ}\)+ sinθ

= sec² θ − 1 + sin θ

Since LHS ≠ RHS, this is not an identity.