Let's simplify the left-hand side of the equation:
cos a - tan a * cos ((3pi/2) - a)
Using the identity cos (90 - x) = sin x, we can write the above expression as:
cos a - tan a * sin a
We know that tan a = sin a / cos a, so substituting this in the equation:
cos a - (sin a / cos a) * sin a
Multiplying through by cos a:
cos^2 a - sin^2 a = (cos^2 a) / cos a
Using the Pythagorean identity sin^2 a + cos^2 a = 1, we have:
1 - sin^2 a - sin^2 a = cos a
2sin^2 a = cos a
Recall that sec a = 1 / cos a. Dividing both sides of the equation by cos a:
sec a = 2sin^2 a / cos a
Using the identity sin^2 a = 1 - cos^2 a:
sec a = 2(1 - cos^2 a) / cos a
sec a = (2 - 2cos^2 a) / cos a
sec a = 2/cos a - 2cos a
Using the identity sec a = 1/cos a:
sec a - 2cos a - 2cos a = 1 / cos a
sec a - 4cos a = 1 / cos a
Multiplying through by cos a:
sec a * cos a - 4cos^2 a = 1
But we already know that cos^2 a - sin^2 a = cos a. Therefore:
sec a * cos a - 4(cos^2 a + sin^2 a) = 1
sec a * cos a - 4cos^2 a - 4sin^2 a = 1
Recall that sin^2 a + cos^2 a = 1. So:
sec a * cos a - 4(cos^2 a + 1 - cos^2 a) = 1
sec a * cos a - 4 = 1
sec a * cos a = 5
Now, we are left with sec a * cos a = 5, which is not equal to the original equation of sec a. Therefore, the original equation is not true for all values of 'a'.
cos a - tan a * cos ((3pi/2) - a) = sec a
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