This question has been taken from the Book- ML Aggarwal
Board- ICSE
Publication- Avichal
Chapter- Trigonometric Identities
Chapter number-18
We have to prove that
(i) (sec A – 1)/(sec A + 1) = (1 – cos A)/(1 + cos A) (ii) tan2 θ/ (sec θ – 1)2 = (1 + cos θ)/ (1 – cos θ) (iii) (1 + tan A)2 + (1 – tan A)2 = 2 sec2 A (iv) sec2 A + cosec2 A = sec2 A. cosec2 A
Class10th, chapter 18, trigonometric identities, ques no. 16
Solution:
(iii) L.H.S. = (1 + tan A)2 + (1 – tan A)2
= 1 + 2 tan A + tan2 A + 1 – 2 tan A + tan2 A
= 2 + 2 tan2 A
= 2(1 + tan2 A) [As 1 + tan2 A = sec2 A]
= 2 sec2 A
= R.H.S.
(iv) L.H.S = sec2 A + cosec2 A
= 1/cos2 A + 1/sin2 A
= (sin2 A + cos2 A)/ (sin2 A cos2 A)
= 1/ (sin2 A cos2 A)
= sec2 A cosec2 A = R.H.S