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 + tan

^{2}A + 1 – 2 tan A + tan^{2}A= 2 + 2 tan

^{2}A= 2(1 + tan

^{2}A) [As 1 + tan^{2}A = sec^{2}A]= 2 sec

^{2}A= R.H.S.

(iv) L.H.S = sec

^{2 }A + cosec^{2}A= 1/cos

^{2}A + 1/sin^{2}A= (sin

^{2}A + cos^{2}A)/ (sin^{2}A cos^{2}A)= 1/ (sin

^{2}A cos^{2}A)= sec

^{2}A cosec^{2}A = R.H.S