Consider the following: 1. tan2 θ – sin2 θ = tan2 θ sin2 θ 2.

1.	Consider the following: 1. tan2 θ – sin2 θ = tan2 θ sin2 θ 2. (1 + cot2 θ) (1– cos θ) (1 + cos θ) = 1Which of the statements given below is correct? (a) 1 only is the identity (b) 2 only is the identity (c) Both 1 and 2 are identities (d) Neither 1 nor 2 is the identity
Answer» 1. tan² θ - sin² θ = \(\frac{sin^2\,\theta}{cos^2\,\theta}\) - sin² θ = \(\frac{sin^2\,\theta - sin^2\,\theta\,cos^2\,\theta}{cos^2\,\theta}\) = \(\frac{sin^2\,\theta(1-cos^2\,\theta)}{cos^2\,\theta}\) = \(\frac{sin^2\,\theta}{cos^2\,\theta}\) x sin² θ = tan² θ sin² θ 2. (1 + cot² θ) ( 1– cos θ) (1 + cos θ) = ( 1+ cot² θ) (1– cos² θ) = \(\bigg(1+\frac{cos^2\,\theta}{sin^2\,\theta}\bigg)(1-cos^2\,\theta)\) = \(\bigg(\frac{sin^2\,\theta + cos^2\,\theta}{sin^2\,\theta}\bigg) \times sin^2\,\theta\) (∴ 1 – cos² θ = sin² θ) = 1. (cos² θ + sin² θ =1) ∴ (c) is the correct option.

Consider the following: 1. tan2 θ – sin2 θ = tan2 θ sin2 θ 2. (1 + cot2 θ) (1– cos θ) (1 + cos θ) = 1Which of the statements given below is correct? (a) 1 only is the identity (b) 2 only is the identity (c) Both 1 and 2 are identities (d) Neither 1 nor 2 is the identity

Answer»

1. tan² θ - sin² θ = \(\frac{sin^2\,\theta}{cos^2\,\theta}\) - sin² θ

= \(\frac{sin^2\,\theta - sin^2\,\theta\,cos^2\,\theta}{cos^2\,\theta}\)

= \(\frac{sin^2\,\theta(1-cos^2\,\theta)}{cos^2\,\theta}\) = \(\frac{sin^2\,\theta}{cos^2\,\theta}\) x sin² θ = tan² θ sin² θ

2. (1 + cot² θ) ( 1– cos θ) (1 + cos θ)

= ( 1+ cot² θ) (1– cos² θ)

= \(\bigg(1+\frac{cos^2\,\theta}{sin^2\,\theta}\bigg)(1-cos^2\,\theta)\)

= \(\bigg(\frac{sin^2\,\theta + cos^2\,\theta}{sin^2\,\theta}\bigg) \times sin^2\,\theta\)

(∴ 1 – cos² θ = sin² θ) = 1. (cos² θ + sin² θ =1)

∴ (c) is the correct option.