$$ 1.\space \frac{1-tan^2 45^\circ}{1+tan^2 45^\circ} \space is \space equal \space to \space: $$

(a) tan 90°

(b) 1

(c) sin 45°

(d) sin 0°

  • Ans. (d) sin 0°
 Explanation :

Given,

310

$$\frac{1-tan^2 45^\circ}{1+tan^2 45^\circ} \space = \space \frac{1-1}{1+1} \space = \space 0 \space = \space sin \space 0^\circ .$$

Q. (1+tan2A)/(1+cot2A) equals to:
(a) sec2 A

(b) – 1

(c) cot2 A
(d) tan2 A
Ans. (d) tan2 A
 Explanation :

Given,

Q. If cos A =4/3, then the value of tan A is:

(a) 3/5

(b) 3/4

(c) 4/3

(b) 5/3

Ans. (b) 3/4

 Explanation :
312
Q. Find the value of 9 sec2 A – 9 tan2 A.

Sol. Given,

9 sec2 A – 9 tan2 A = 9(sec2 A – tan2 A)
= 9(1) [... sec2 q – tan2 q = 1]

= 9. Ans.

Q. Find the value of (1 + tan θ + sec θ) (1 + cot θ – cosec θ)

Sol. Given,

  • (1 + tan θ + sec θ ) (1 + cot θ – cosec θ )
  • = [1 + tan θ + sec θ + cot θ + tan θ cot θ + sec θ cot θ – cosec θ – tan θ cosec θ – sec θ cosec θ ]
  • = [1 + tan θ + sec θ + cot θ + 1 + cosec θ – cosec θ – sec θ – sec θ cosec θ ]
  • = [2 + tan θ + cot θ – sec θ cosec θ ]
  • = 2 + sin θ/Cos θ + cos θ/sin θ - 1/sin θ cos θ
  • = 2 + ((sin 2 θ + cos 2 θ)-1)/sin θ cos θ
  • = 2+((1-1)/sin θ cos θ) [... sin2 θ + cos2 θ = 1]
  • = 2

Q. Find (sec A + tan A) (1 – sin A).

Sol. Given,

Q. In ∆OPQ right angled at P, OP = 7 cm , OQ – PQ = 1 cm. Determine the values of sin Q and cos Q.

Sol. In ∆OPQ, we have

  • OQ2 = OP2 + PQ2
  • ⇒ (PQ + 1)2 = OP2 + PQ2
  • [·.· OQ – PQ = 1 ⇒ OQ = 1 + PQ]
314
  • ⇒ PQ2 + 2PQ + 1 = OP2 + PQ2
  • ⇒ 2PQ + 1 = 72
  • ⇒ 2PQ + 1 = 49
  • ⇒ 2PQ = 48
  • ⇒ PQ = 24 cm
  • and OQ – PQ = 1 cm
  • OQ = (PQ + 1) cm
  • = 25 cm
  • Thus, sin Q =QP/OQ= 7/25
  • and cos Q =PQ/OQ=24/25 Ans.

Q. If ∠B and ∠Q are acute angles such that sin B = sin Q, then prove that ∠B = ∠Q.

  • Sol. Consider two right angled triangles ABC and PQR such that sin B = sin Q.
  • We have, sin B = AC/AB and sin Q =PR/PQ
315
316
Q. In ∆ABC right – angled at C, AB = 29 units, BC = 21 units and ∠ABC = θ. Determine the values of (i) cos2 θ + sin2 θ, and (ii) cos2 θ – sin2 θ.
  • Ans. (i) 1, (ii)41/841

Q. In ∆OPQ, right-angled at P, OP = 7 cm, OQ – PQ = 1 cm. Determine the values of sin Q and
cos Q.

  • Ans. sin Q =7/25, cos Q = 24/25

Q. In ∆PQR, right-angled at Q, PQ = 5 cm, PR + QR = 25 cm. Determine the values of sin P, cos P and tan P.

  • Ans. sin P =12/13, cos P =5/13 and tan P = 12/5.

Q. In ∆ABC, right-angled at B, if tan A =1/√3, find the value of (i) sin A cos C + cos A sin C, and (ii) cos A cos C – sin A sin C.

  • Ans. (i) sin A cos C + cos A sin C = 1
  • (ii) cos A cos C – sin A cos C = 0.

Q. In a ∆ABC, right-angled at B, AB = 24 cm, BC = 7 cm. Determine (i) sin A, cos A and (ii) sin C, cos C.

  • Ans. (i) sin A = 7/25 and cos A = 24/25
  • (ii) sin C = 24/25 and cos C = 7/25

Q. Given 15 cot A = 8. Find sin A and sec A.

  • Ans. sin A = 15/17, sec A = 17/8·
Q. If cot θ = 7/8 evaluate (i) (1+sin θ)(1-sin θ)/(1+cos θ)(1-cos θ) and (ii) cot2 θ.
  • Ans. (i) 49/64, (ii) 49/64·
Q. If 3 cot A = 4, check whether (1-tan2A)/(1+tan2A) = cos2 A – sin2 A or not.
  • Ans. Both are equal to 7/25·

Q. If tan (A – B) = 1/√3 and tan (A + B) = √3, 0° < A + B ≤ 90° and A > B, find A and B.

  • Ans. A = 45° and B = 15°.

Q. If sin (A – B) = 1/2 and cos (A + B) = 1/2, 0° < A + B ≤ 90° and A > B, find A and B.

  • Ans. A = 45° and B = 15°.

Q. In ∆PQR, right-angled at Q, PQ = 3 cm and PR = 6 cm. Determine ∠P and ∠R.

  • Ans. ∠P = 60° and ∠R = 30°.