NCERT Solutions
Class 10 Maths
Introduction to Trigonometry
Ex.8.3 Q.3
Choose the correct option. Justify your choice.
(1) 9 sec2 A – 9 tan2 A =
(A) 1
(B) 9
(C) 8
(D) 0
(2) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =
(A) 0
(B) 1
(C) 2
(D) -1
(3) (sec A + tan A) (1 – sin A) =
(A) sec A
(B) sin A
(C) cosec A
(D) cos A
(4) (1 + tan2 A) ÷ (1 + cot2 A) =
(A) sec2 A
(B) –1
(C) cot2 A
(D) tan2 A
(1) 9 sec2 A – 9 tan2 A =
Given, 9 sec2 A - 9 tan2 A
= 9(sec2 A - tan2 A)
= 9 × 1
{since sec2 A - tan2 A = 1}
= 9
Hence, option (B) is the correct answer.
(2) (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =
= 1 + cot θ – cosec θ + tan θ + tan θ cot θ – tan θ cosec θ + sec θ + sec θ cot θ - sec θ cosec θ
= 1 + 
– 
+ 
+ 1 – × + 
+ × – ×
[since tan θ cot θ = 1]
= 2 + – +
– + + – × 
= 2 + + –
= 2 + (cos2 θ + sin2 θ - 1) ÷ (cos θ × sin θ)
= 2 + [(1 - 1) ÷ (cos θ * sin θ)]
[sin2 θ + cos2 θ = 1]
= 2 + 0
= 2
Hence, option (C) is the correct answer.
(3) (sec A + tan A) (1 – sin A) = (
+
) (1 – sin A)
= {(
}(1 – sin A)
= (1 – sin2 A) ÷ cos A
= cos2 A ÷ cos A
[sin2 θ + cos2 θ = 1]
= cos A
Hence, option (D) is the correct answer.
(4) (1 + tan2 A) ÷ (1 + cot2 A) = sec2 A ÷ cosec2 A
[sec2 A = 1 + cot2 A and cosec2 A = 1 + tan2 A]
= (
) ÷ (
)
= 
= tan2 A
Hence, option (D) is the correct answer.