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

sin²a = (5 cos a)²

= 25 cos²a

so, sin²a = 25 cos²a, name this as equation1

using sin²a + cos²a = 1, name this as equation2

substitute equation1 into equation2

25 cos²a + cos²a = 1

26 cos²a = 1

cos²a = 1/26

using the double angle formula

cos2a = 2 cos²a - 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.