1 – 2 sin²θ + sin⁴ θ
= (1 – sin²θ)² = (cos²θ)² = cos⁴θ

Correct Option: B

1 – 2 sin²θ + sin⁴ θ
= (1 – sin²θ)² = (cos²θ)² = cos⁴θ

Expression
= cot 9°. cot 27°. cot 63° . cot 81°
= cot 9°. cot 27°. cot (90° – 27°) . cot (90° – 9°)
= cot 9° . cot 27° . tan 27° . tan 9° [tan (90° – θ)
= cot θ; cot (90° – θ) = tan θ ]
= cot 9° . tan 9° . cot 27° tan 27°
= 1 [tan θ . cot θ = 1]

Correct Option: B

Expression
= cot 9°. cot 27°. cot 63° . cot 81°
= cot 9°. cot 27°. cot (90° – 27°) . cot (90° – 9°)
= cot 9° . cot 27° . tan 27° . tan 9° [tan (90° – θ)
= cot θ; cot (90° – θ) = tan θ ]
= cot 9° . tan 9° . cot 27° tan 27°
= 1 [tan θ . cot θ = 1]

(1 + sin A) (1 + sin B) (1 + sin C) = (1 – sin A) . (1 – sin B) (1 – sin C) = x (Let)
∴ x . x = (1 + sin A) (1 + sin B) (1 + sin C) (1 – sin A) (1 – sin B) (1 – sin C)
⇒ x² = (1 – sin² A) (1 – sin² B) (1 – sin² C)
⇒ x² = cos²A . cos²B . cos²C
⇒x = ± cos A . cos B . cos C

Correct Option: B

(1 + sin A) (1 + sin B) (1 + sin C) = (1 – sin A) . (1 – sin B) (1 – sin C) = x (Let)
∴ x . x = (1 + sin A) (1 + sin B) (1 + sin C) (1 – sin A) (1 – sin B) (1 – sin C)
⇒ x² = (1 – sin² A) (1 – sin² B) (1 – sin² C)
⇒ x² = cos²A . cos²B . cos²C
⇒x = ± cos A . cos B . cos C

tan²θ + 3 = 3 sec θ
⇒ sec²θ – 1 + 3 = 3 sec θ
⇒ sec²θ – 3 sec θ + 2 = 0
⇒ sec²θ – 2 sec θ – sec θ + 2 = 0
⇒ secθ (sec θ – 2) – 1 (sec θ – 2) = 0
⇒ (sec θ – 2) (sec θ – 1) = 0
⇒ secθ = 2 or 1
⇒ θ = 60° or 0°.

Correct Option: D

tan²θ + 3 = 3 sec θ
⇒ sec²θ – 1 + 3 = 3 sec θ
⇒ sec²θ – 3 sec θ + 2 = 0
⇒ sec²θ – 2 sec θ – sec θ + 2 = 0
⇒ secθ (sec θ – 2) – 1 (sec θ – 2) = 0
⇒ (sec θ – 2) (sec θ – 1) = 0
⇒ secθ = 2 or 1
⇒ θ = 60° or 0°.

sin θ = 0.7
∴ cos θ
= √1 - sin² θ= √1 - (0.7)²
= √1 - 0.49= √0.51

Correct Option: C

sin θ = 0.7
∴ cos θ
= √1 - sin² θ= √1 - (0.7)²
= √1 - 0.49= √0.51