26. To prove that sec2 A

With the figure of the previous article, we have

1+tan2 A.

Example. Prove that cos A√sec2 4 – 1=sin 4.

cos Asec A-1=cos 4 × tan A

Example. Prove that cot a -1=cosec1 a -2 cosec2 a.

cot1a-1= (cot2 a + 1) (cot2 a − 1)

=cosec2 a (cosec2 a -1 -1)

28. The formulæ proved in the last three articles are not independent, for they are merely different ways of expressing in trigonometrical symbols the property of a right-angled triangle proved in Euc. I. 47.

29. It will be useful here to collect the formulæ proved in this chapter.

30. We shall now exemplify the use of these fundamental formulæ in proving identities. An identity asserts that two expressions are always equal, and the proof of this equality is called "proving the identity." Some easy illustrations have already been given in this chapter. The general method of procedure is to choose one of the expressions given (usually the more complicated of the two) and to shew by successive transformations that it can be made to assume the form of the

other.

Example 1. Prove that sin2 A cot2 A+ cos2 A tana A=1.

Here it will be found convenient to express all the trigonometrical ratios in terms of the sine and cosine.

The first side = sin2 A.

cos2 A
sin2 A

+ cos2 A.

sin2 A cos2 A

=cos2A + sin2 A

Example 2. Prove that sec10-sec2 0 = tan2 0 + tan1 0.

The form of this identity at once suggests that we should use the secant-tangent formula of Art. 26; hence

3. cot A sec A =cosec A.

5. cos A cosec A=cot A. 7. (1-cos2 A) cosec2 A = 1. 8. (1-sin2 A) sec2 A=1. 9. cot20 (1-cos2 ) = cos2 0. 10. (1-cos20) sec2 0=tan2 0. 11. tan a 1-sin2 a =sin a. 12. cosec a √/1-sin2 a = cot a. 13. (1+tan2 A) cos2 A = 1. 15. (1-cos2 ) (1+tan2 6)=tan2 0. 16. cos a cosec a √/sec2 a — 1=1. 17. sin2 A (1+cot2 A)=1.

19. (1-cos2 A) (1+cot2 A)=1.

20. sin a sec a √cosec2 a − 1 = 1.

cos A tan Asin A.

4. sin A sec A=tan A. 6. cot A sec A sin A = 1.

14.

(sec2 A-1) cot2 A = 1.

18. (cosec2 A − 1) tan2 A = 1.

21. cos a √cot2a+1=√/cosec2 a— 1. 22. sin2 cot20+ sin2 0=1.

23. (1+tan2 0) (1 − sin2 0) = 1.

24. sin2 0 sec2 0=sec20-1. 25.

cosec20 tan20-1=tan2 0.

H. K. E. T.

2

30. sin1a-cos1 a= 2 sin2a-1=1-2 cos2 a.

31. sec1 a-1=2 tan2 a+tan* a.

32. cosec1 a-1=2 cot2 a+cot1a.

33. (tan a cosec a)2 — (sin a sec a)2 = 1.

34. (sec cot 0)2 - (cos e cosec )2 = 1. 35. tan20-cot20=sec2 0 - cosec2 0.

31. The foregoing examples have required little more than a direct application of the fundamental formula; we shall now give some identities offering a greater variety of treatment.

Example 1. Prove that sec2 4+cosec2 A = sec2 A cosec2 A.

Occasionally it is found convenient to prove the equality of the two expressions by reducing each to the same form.

Example 2. Prove that

sin2 A tan A+ cos2 A cot A +2 sin A cos Atan A+ cot A.

The transformations in the successive steps are usually suggested by the form into which we wish to bring the result. For instance, in this last example we might have proved the identity by substituting for the tangent and cotangent in terms of the sine and cosine. This however is not the best method, for the form in which the right-hand side is given suggests that we should retain tan a and cot ẞ unchanged throughout the work.

7. √1+cot2 A.√/sec2 A-1.√/1 — sin2 A = 1.

10. (cot 0-1)2+(cot 0+1)2 = 2 cosec2 0.

11. sin2 A (1+cot2 A) + cos2 A (1 + tan2 A) = 2.

12. cos2 A (sec2 A − tan2 A)+sin2 A (cosec2 A – cot2 A)=1.