TOPICS
Exercise - 8.4

Question-1 :-  Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.

Solution :-
  tan A = 1/cot A
  sec A =  √1 + tan² A
        =  √1 + 1/cot² A
        =  √(cot² A + 1)/cot A
  sin A = 1/cosec A = 1/√1 + cot² A
    

Question-2 :-  Write all the other trigonometric ratios of ∠ A in terms of sec A.

Solution :-
  cos A = 1/sec A
  tan A = √sec² A - 1
  cot A = 1/tan A = 1/√sec² A - 1
  sin A = √1 - cos² A
        = √1 - 1/sec² A
        = √(sec² A - 1)/sec² A
        = √(sec² A - 1)/sec A
  cosec A = 1/sin A = sec A/√sec² A - 1
    

Question-3 :-  Evaluate : (i)trigonometry
(ii) sin 25° cos 65° + cos 25° sin 65°

Solution :-
(i)trigonometry
(ii) sin 25° cos 65° + cos 25° sin 65° 
=  sin (90° - 65°) cos 65° + cos (90° - 65°) sin 65° 
=  cos 65° . cos 65° + sin 65° . sin 65°
=  cos² 65° + sin² 65°     [sin² A + cos² A = 1]
=  1
    

Question-4 :-  Choose the correct option. Justify your choice.
(i) 9 sec² A – 9 tan² A =
(A) 1  (B) 9  (C) 8  (D) 0

Solution :-
  9 sec² A – 9 tan² A 
= 9(1 + tan² A) - 9 tan² A 
= 9 + 9 tan² A - 9 tan² A 
= 9
  So, Option B is correct Answer.
    

(ii)  (1 + tan θ + sec θ) (1 + cot θ – cosec θ) =
(A) 0  (B) 1  (C) 2  (D) –1

Solution :-
        trigonometry
  So, Option C is correct Answer.
    

(iii)  (sec A + tan A) (1 – sin A) =
(A) sec A  (B) sin A  (C) cosec A  (D) cos A

Solution :-
        trigonometry
  So, Option D is correct Answer.
    

(iv)  trigonometry
(A) sec² A  (B) –1  (C) cot² A  (D) tan² A

Solution :-
trigonometry
  So, Option D is correct Answer.
    

Question-5 :-  Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(i) trigonometry Solution :-

trigonometry
    

(ii)  trigonometry

Solution :-
trigonometry
    

(iii) trigonometry

Solution :-
trigonometry
    

(iv)  trigonometry

Solution :-
trigonometry
    

(v)  trigonometry

Solution :-
        trigonometry
    

(vi) trigonometry

Solution :-
        trigonometry
    

(vii)  trigonometry

Solution :-
        trigonometry
    

(viii)  trigonometry

Solution :-
trigonometry
    

(xi) trigonometry

Solution :-
        trigonometry
    

(x)  trigonometry

Solution :-
trigonometry
    
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