# 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
5 years, 5 months ago

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

$(\sin \theta + \cos \theta)(\tan \theta + \cot \theta)=\sec \theta + \csc \theta$

- 5 years, 5 months ago

Can you hear the Proving Trigonometric Identities Wiki page calling out your name?

Staff - 5 years, 5 months ago

Thank You, it is a good practice for we beginners!

- 5 years, 5 months ago

yes

- 2 years, 7 months ago