# What is tan2a if cosa = 1 - sina?

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cosa = 1-sina.

To find tan2a.

cosa = 1-sina

sqrt(1-sin^2a) = 1-sina, as cos^2x+sin^2x = 1.

Squaring both sides , we get:

1-sin^2a = (1-sina)^2

0 = (1-sina)^2 - (1-sin^2a)

0 - (1-sina)(1-sina +1+sina)

0 = (1-sina)2

sina = 1.

Therefore a = pi/2.

cosa = cos pi/2 = 0.

Therefore tan 2a = tan (2*pi/2) = tan pi = 0

Therefore tan 2a = 0.

We'll re-write the given constraint cosa = 1 - sina. We'll add sin a both sides and we'll have:

sin a + cos a = 1

We'll square raise the new relation sina + cosa = 1.

(sina + cosa)^2 = 1^2

(sina)^2 + (cosa)^2 + 2sina*cosa = 1 (1)

But, from the fundamental formula of trigonometry:

(sina)^2 + (cosa)^2 = 1

We'll substitute (sina)^2 + (cosa)^2 by 1:

The relation (1) will become:

1 + 2sina*cosa = 1

We'll eliminate like terms:

2sina*cosa = 0

But 2sina*cosa = sin (2a)

We'll write the formula for tan 2a:

tan 2a = sin 2a/cos 2a

tan 2a = 0/cos 2a

**tan 2a = 0**