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Trigonometry practice for beginners

The following problems need only trigonometric definition of functions and the equation

$\sin^2{\theta} + \cos^2{\theta} = 1$

unless otherwise stated. More will be written.

Prove the following:

1. $$\csc{\theta} \cdot \cos{\theta} = \cot{\theta}$$

2. $$\csc{\theta}\cdot\tan{\theta}=\sec{\theta}$$

3. $$1+\tan^2(-\theta) = \sec^2{\theta}$$

4. $$\cos{\theta}(\tan{\theta}+\cot{\theta})=\csc{\theta}$$

5. $$\sin{\theta}(\tan{\theta}+\cot{\theta})=\sec{\theta}$$

6. $$\tan{\theta}\cdot\cot{\theta}-\cos^2{\theta}=\sin^2{\theta}$$

7. $$\sin{\theta}\cdot\csc{\theta}-\cos^2{\theta}=\sin^2{\theta}$$

8. $$(\sec{\theta}-1)(\sec{\theta}+1)=\tan^2{\theta}$$

9. $$(\csc{\theta}-1)(\csc{\theta}+1)=\cot^2{\theta}$$

10. $$(\sec{\theta}-\tan{\theta})(\sec{\theta}+\tan{\theta})=1$$

11. $$\sin^2{\theta}(1+\cot^2{\theta})=1$$

12. $$(1-\sin^2{\theta})(1+\tan{\theta})=1$$

13. $$(\sin{\theta}+\cos{\theta})^2+(\sin{\theta}-\cos{\theta})=2$$

14. $$\tan^2{\theta}\cdot\cos^2{\theta}+\cot^2{\theta}\cdot\sin^2{\theta}=1$$

15. $$\sec^4{\theta}-\sec^2{\theta}=\tan^4{\theta}+\tan^2{\theta}$$

16. $$\sec{\theta}-\tan{\theta}=\dfrac{\cos{\theta}}{1+\sin{\theta}}$$

17. $$\csc{\theta}-\cot{\theta}=\dfrac{\sin{\theta}}{1+\sin{\theta}}$$

18. $$3\sin^2{\theta}+4\cos^2{\theta}=3+\cos^2{\theta}$$

19. $$9\sec^2{\theta}-5\tan^2{\theta}=5+4\sec^2{\theta}$$

20. $$1-\dfrac{\cos^2{\theta}}{1+\sin{\theta}}=\sin{\theta}$$

21. $$1-\dfrac{\sin^2{\theta}}{1-\cos{\theta}}=-\cos{\theta}$$

22. $$\dfrac{1+\tan{\theta}}{1-\tan{\theta}}=\dfrac{\cot{\theta}+1}{\cot{\theta}-1}$$

23. $$\dfrac{\sec{\theta}}{\csc{\theta}}+\dfrac{\sin{\theta}}{\cos{\theta}}=2\tan{\theta}$$

24. $$\dfrac{\csc{\theta}-1}{\cot{\theta}}=\dfrac{\cot{\theta}}{\csc{\theta}+1}$$

25. $$\dfrac{1+\sin{\theta}}{1-\sin{\theta}}=\dfrac{\csc{\theta}+1}{\csc{\theta}-1}$$

26. $$\dfrac{1-\sin{\theta}}{\cos{\theta}}+\dfrac{\cos{\theta}}{1-\sin{\theta}}=2\sec{\theta}$$

27. $$\dfrac{\sin{\theta}}{\sin{\theta}-\cos{\theta}}=\dfrac{1}{1-\cot{\theta}}$$

28. $$1-\dfrac{\sin^2{\theta}}{1+\cos{\theta}}=\cos{\theta}$$

29. $$\dfrac{1-\sin{\theta}}{1+\sin{\theta}}=(\sec{\theta}-\tan{\theta})^2$$

30. $$\dfrac{\cos{\theta}}{1-\tan{\theta}}+\dfrac{\sin{\theta}}{1-\cot{\theta}}=\sin{\theta}+\cos{\theta}$$

31. $$\dfrac{\cot{\theta}}{1-\tan{\theta}}+\dfrac{\tan{\theta}}{1-\cot{\theta}}=1+\tan{\theta}+\cot{\theta}$$

32. $$\tan{\theta}+\dfrac{\cos{\theta}}{1+\sin{\theta}}=\sec{\theta}$$

Note by Sharky Kesa
1 year, 5 months ago

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Here's a nice result which involves all six ratios :

$(\sin \theta + \cos \theta)(\tan \theta + \cot \theta)=\sec \theta + \csc \theta$ · 1 year, 5 months ago

Can you hear the Proving Trigonometric Identities Wiki page calling out your name? Staff · 1 year, 5 months ago