find the value of (cos a - sin a) / (cos a + sin a )

Question

mathslover · Answer

Multiply the conjugate of "cos a + sin a " to both numerator and denominator. 

Can you tell me what is the conjugate of "cos a + sin a" ?

anonymous · Answer

Like @mathslover rightly said, multiply up and bottom by conjugate of denominator $(cosa-sina)$

$\dfrac{(cosa-sina)(cosa-sina)}{cosa + sina)(cosa-sina)} = \dfrac{cos^2a -2sinacosa-sin^2a}{cos^2a+sin^2a}$

Now, recall the trig identities:

$cos^2a -sin^2a = cos2a, cos^2a + sin^2a = 1 $, and $2sinacosa = sin2a$

Substitute to get:

$sin2a + cos2a$

anonymous · Answer

Let me know if something isn't quite clear

anonymous · Answer

Oh no, wait. Think I got the signs mixed up somewhere...

anonymous · Answer

$\dfrac{cos^2a -2sinacosa+sin^2a}{cos^2a-sin^2a} \longrightarrow 
\dfrac{cos^2a +sin^2a -2sinacosa}{cos2a} = \dfrac{1 - sin2a}{cos2a}$

anonymous · Answer

You can further simplify that to $\dfrac{1}{cos2a} - \dfrac{sin2a}{cos2a} \longrightarrow sec2a - tan2a$