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### How to find cos2a from tan a?

Question:

If tan a = 5, what is the value of cos 2a?

Solution:

- by definition: tan a = sin a/cos a
- given: tan a = 5, then sin a/cos a = 5
- so, sin a = 5 cos a
- square both sides of the equation
- sin
^{2}a = (5 cos a)^{2} - = 25 cos
^{2}a - so, sin
^{2}a = 25 cos^{2}a, name this as equation1 - using sin
^{2}a + cos^{2}a = 1, name this as equation2 - substitute equation1 into equation2
- 25 cos
^{2}a + cos^{2}a = 1 - 26 cos
^{2}a = 1 - cos
^{2}a = 1/26 - using the double angle formula
- cos2a = 2 cos
^{2}a - 1 - = 2 × (1/26) - 1
- = 2/26 - 1
- =1/13 - 1
- =1/13 - 13/13
- = - 12/13

Therefore, the value of cos 2a is -12/13. Watch the video for more details.

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