Question

If `tan theta = (sin alpha - cos alpha) /(sin alpha + cos alpha)`, then `sin alpha+cos alpha` is

Answer 1

`tan theta = ( sin alpha/cos alpha - 1)/(sin alpha/cos alpha + 1)`
`tan theta = ( tan alpha - 1)/(tan alpha + 1)`
`tan theta = ( tan alpha - tan (pi/4))/( 1 + tan alpha* tan(pi/4))`
`tan theta = tan(alpha - pi/4)`
`theta = alpha - pi/4`
`alpha = theta + pi/4`
now,`sin(theta+ pi/4) + cos(theta + pi/4)`
`= sin theta 1/sqrt2 + cos theta 1/sqrt2 + 1/sqrt2 cos theta - 1/sqrt2 sin theta`
`= 2*1/sqrt2 cos theta`
`= sqrt2 cos theta`
option 2 is correct