Choose the correct option. Justify your choice.
(i) 9 sec2A - 9 tan2A =
(A) 1 (B) 9 (C) 8 (D) 0
(ii) (1 + tan θ + sec θ) (1 + cot θ - cosec θ)
(A) 0 (B) 1 (C) 2 (D) - 1
(iii) (secA + tanA) (1 - sinA) =
(A) secA (B) sinA (C) cosecA (D) cosA
(iv) 1+tan2A/1+cot2A =
(A) sec2A (B) -1 (C) cot2A (D) tan2A
(i) (B) is correct.
9 sec2A - 9 tan2A
= 9 (sec2A - tan2A)
= 9×1 = 9 (��� sec2 A - tan2 A = 1)
(ii) (C) is correct
(1 + tan θ + sec θ) (1 + cot θ - cosec θ)
= (1 + sin θ/cos θ + 1/cos θ) (1 + cos θ/sin θ - 1/sin θ)
= (cos θ+sin θ+1)/cos θ × (sin θ+cos θ-1)/sin θ
= (cos θ+sin θ)2-12/(cos θ sin θ)
= (cos2θ + sin2θ + 2cos θ sin θ -1)/(cos θ sin θ)
= (1+ 2cos θ sin θ -1)/(cos θ sin θ)
= (2cos θ sin θ)/(cos θ sin θ) = 2
(iii) (D) is correct.
(secA + tanA) (1 - sinA)
= (1/cos A + sin A/cos A) (1 - sinA)
= (1+sin A/cos A) (1 - sinA)
= (1 - sin2A)/cos A
= cos2A/cos A = cos A
(iv) (D) is correct.
1+tan2A/1+cot2A
= (1+1/cot2A)/1+cot2A
= (cot2A+1/cot2A)×(1/1+cot2A)
= 1/cot2A = tan2A
