Choose the correct option. Justify your choice.
(i) 9 sec²A - 9 tan²A =
(A) 1   (B) 9   (C) 8   (D) 0
(ii) (1 + tanθ + secθ) (1 + cotθ - cosecθ) =
(A) 0   (B) 1   (C) 2   (D) -1
(iii) (sec A + tan A) (1 - sin A) = 
(A) sec A    (B) sin A    (C) cosec A    (D) cos A
(iv) (1 + tan² A)/(1 + cot² A) =
(A) sec²A   (B) -1   (C) cot²A   (D) tan²A

Solution:

We will use the basic trigonometric identities and properties of the trigonometric ratios to solve the problem.

sin² A + cos² A = 1
cosec² A = 1 + cot² A
sec² A = 1 + tan² A

(i) 9 sec²A - 9 tan²A

= 9 (sec² A - tan² A)

= 9 × 1 [By using the identity, 1 + sec² A = tan² A]

= 9

Thus, option (B) is the correct answer.

(ii) (1 + tanθ + secθ) (1 + cotθ - cosecθ) -------- (1)

We know that using the trigonometric ratios,

tan (x) = sin (x)/cos (x)

cot (x) = cos (x)/sin(x) = 1/tan (x)

sec (x) = 1/cos (x)

cosec (x) = 1/sin (x)

By substituting the above function in equation (1),

= [(1 + sinθ/cosθ + 1/cosθ) (1 + cosθ/sinθ - 1/sinθ)]

= [(cosθ + sinθ + 1)/cosθ] [(sinθ + cosθ - 1)/sinθ] (By taking LCM and multiplying)

= [(sinθ + cosθ)² - (- 1)²] / sinθ cosθ [Using a² - b² = (a + b)(a - b)]

= [sin² θ + cos² θ + 2 sinθ cosθ - 1] / sinθ cosθ

= (1 + 2 sinθ cosθ - 1) / sinθ cosθ (Using identity sin² θ + cos² θ = 1)

= 2 sinθ cosθ / sinθ cosθ

= 2

Hence, option (C) is correct.

(iii) (sec A + tan A) (1 - sin A)

We know that,

tan(x) = sin(x) / cos(x)

sec(x) = 1/cos(x)

By substituting these values in the given expression we get, 

= [(1/cos A) + (sin A/cos A)] (1 - sin A)

= [(1 + sin A)/cos A] (1 - sin A)

= (1 - sin² A) / cos A

= cos² A/cos A (By using the identity sin²θ + cos²θ = 1)

= cos A

Hence, option (D) is correct.

(iv) (1 + tan² A) / (1 + cot² A)

We know that,

tan (x) = sin (x) / cos (x)

cot (x) = cos (x) / sin (x)

= 1/tan (x)

By substituting these in the given expression we get,

(1 + tan² A) / (1 + cot² A) = (1 + sin² A/cos² A) / (1 + cos² A/sin² A)

= [(cos² A + sin² A)/cos² A] [(sin² A + cos² A)/sin² A]

 = (1/cos² A)/(1/sin² A) [By using the identity: sin² A + cos² A = 1]

= sin² A/cos² A

= tan² A

Hence, option (D) is correct.

☛ Check: NCERT Solutions Class 10 Maths Chapter 8

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Video Solution:

Choose the correct option. Justify your choice. (i) 9 sec²A - 9 tan²A = (A) 1   (B) 9   (C) 8   (D) 0 (ii) (1 + tanθ + secθ) (1 + cotθ - cosecθ) = (A) 0   (B) 1   (C) 2   (D) -1 (iii) (sec A + tan A) (1 - sin A) = (A) sec A    (B) sin A    (C) cosec A    (D) cos A (iv) (1 + tan²A)/(1 + cot²A) = (A) sec²A   (B) -1   (C) cot²A   (D) tan²A

Maths NCERT Solutions Class 10 Chapter 8 Exercise 8.4 Question 4

Summary:

The correct answers for the following 9 sec²A - 9 tan²A, (1 + tan θ + sec θ)(1 + cot θ − cosec θ), (sec A + tan A) (1 -sin A) and (1 + tan² A)/(1 + cot² A) are: (B) 9, (C) 2, (D) cos A and (D) tan² A, respectively.

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☛ Related Questions:

• Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.
• Write all the other trigonometric ratios of ∠A in terms of sec A.
• Evaluate:(i) (sin² 63° + sin² 27) / (cos² 17° + cos² 73°)(ii) sin 25° cos 65° + cos 25° sin 65°
• Prove the following identities, where the angles involved are acute angles for which the expressions are defined.(i) (cosecθ - cotθ)² = 1 - cosθ/1 + cosθ(ii) cos A/(1 + sin A) + (1 + sin A)/cos A = 2 sec A(iii) tanθ/(1 - cotθ) + cotθ/(1 - tanθ) = 1 + secθ cosecθ(iv) (1 + sec A)/sec A = sin² A/(1 - cos A)(v) (cos A - sin A + 1)/cos A + sin A + 1 = cosec A + cot A(vi) 1 + sin A/(1 - sin A) = sec A+ tan A(vii) (sinθ - 2sin³ θ)/(2cosθ - cosθ) = tanθ(viii) (sin A + cosec A)² + (cos A + sec A)² = 7 + tan² A + cot² A(ix) (cosec A - sin A)(sec A - cos A) = 1/(tan A + cot A)(x) [(1 + tan² A)/(1 + cot² A)] = [(1 - tan A/1 - cot A)²] = tan² A