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1 (1) Veriﬁcation of trigonometric identities Fundamental Reciprocal identities sinx = 1 cscx ; cosx = 1 secx ; tanx = 1 cotx Ratio identities tanx = sinx cosx ; cotx = cosx sinx Pythagorean identities sin 2 x + cos 2 x = 1; 1 + cot 2 x = sec 2 x; tan 2 x + 1 = csc 2 x . 1. Chang e to sines and cosins to verif y an Iden tity sinx cotx sec x = 1 2. Use pythag orean iden tity to ver ify an inden tity 1 − 2sin 2 x = 2cos 2 x − 1 3. F actor to verify an indentit y csc 2 x − cos 2 x csc 2 x = 1 4. Multiply by a conju gate to verify an identi ty sinx 1 + cosx = 1 − cosx sinx 5. Chang e to sines and cosine s to ver ify an indentit y sinx + tanx 1 + cosx = tanx Exercises 1. tanx csc x cosx = 1 (1) 2. tanx sec x sinx = tan 2 x (1) 3. 4sin 2 x − 1 2sinx + 1 = 2sinx − 1 (3) 4. sin 2 x − 2sinx + 1 sinx − 1 = sinx − 1 5. (sinx − cosx)(sinx + cosx) = 1 − 2cos 2 x 6. (tanx)(1 − cotx) = tanx − 1 7. 1 sinx − 1 cosx = cosx − sinx sinx cosx 8. 1 sinx + 1 cosx = cosx + 3sinx sinx cosx 9. cosx 1 − sinx = secx + tanx 10. sinx 1 − cosx = cscx + cotx 11. 1 − tan 4 x sec 2 x = 1 − tan 2 x 12. sin 4 x − cos 4 = sin 2 x − cos 2 x 13. 1 − tan 3 x 1 + tanx = 1 − tanx + tan 2 x 14. cosx tanx − sinx cotx = 0 15. sinx − 2 + 1 sinx sinx − 1 sinx = sinx − 1 sinx + 1

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identidades trigonometricas

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1

(1)

Verification of trigonometric identities

Fundamental Reciprocal identities sinx =1

cscx; cosx =

1

secx; tanx =

1

cotx

Ratio identities tanx =sinx

cosx; cotx =

cosx

sinx

Pythagorean identities sin2x+ cos2x = 1; 1 + cot2x = sec2x; tan2x+ 1 = csc2x.

1. Change to sines and cosins to verify an Identitysinx cotx secx = 1

2. Use pythagorean identity to verify an indentity1 2sin2x = 2cos2x 1

3. Factor to verify an indentitycsc2x cos2x csc2x = 1

4. Multiply by a conjugate to verify an identitysinx

1 + cosx=

1 cosxsinx

5. Change to sines and cosines to verify an indentitysinx+ tanx

1 + cosx= tanx

Exercises

1. tanx cscx cosx = 1 (1)

2. tanx secx sinx = tan2x (1)

3.4sin2x 12sinx+ 1

= 2sinx 1 (3)

4.sin2x 2sinx+ 1

sinx 1= sinx 1

5. (sinx cosx)(sinx+ cosx) = 1 2cos2x

6. (tanx)(1 cotx) = tanx 1

7.1

sinx 1

cosx=

cosx sinxsinx cosx

8.1

sinx+

1

cosx=

cosx+ 3sinx

sinx cosx

9.cosx

1 sinx= secx+ tanx

10.sinx

1 cosx= cscx+ cotx

11.1 tan4xsec2x

= 1 tan2x

12. sin4x cos4 = sin2x cos2x

13.1 tan3x1 + tanx

= 1 tanx+ tan2x

14.cosx tanx sinx

cotx= 0

15.sinx 2 + 1

sinx

sinx 1sinx

=sinx 1sinx+ 1