General solution of a Trigonometric Equation :
(i) If `sin theta = sin alpha => theta = n pi+ (-1)^n alpha`, where ` alpha in [-pi/2 , pi/2 ] , n in I`.
(ii) If `cos theta = cos alpha => theta = 2 n pi pm alpha`, where ` alpha in [0, pi], n on I`
(iii) If `tan theta = tan alpha => theta = n pi + alpha ` , where `alpha in (-pi/2, pi/2), n in I`
(iv) If `sin^2 theta = sin^2 alpha => theta = n pi pm alpha `
(v) lf `cos^2 theta = cos^2 alpha => n pi pm alpha`
(vi) lf `tan^2 theta = tan^2 alpha => theta = n pi pm alpha`
Note : `alpha` is called the principal angle.
Proof :
(i) `sin theta = sin alpha => sin theta - sin alpha =0 => 2 cos ((theta+alpha)/2) sin ((theta - alpha)/2) =0`
`=> cos ((theta+alpha)/2) =0 ` or `sin ( (theta - alpha)/2)=0`
`=> (theta+ alpha)/2 =(2m+1) pi/2 ` or `(theta - alpha)/2 = m pi`, where `m in I`
`=> theta = (2m +1) pi - alpha ` or `theta = 2m pi + alpha` , where `m in I`
`=> theta = (2m+1) pi +(-1)^(2m+1) alpha ` or `theta =2 m pi+ (-1)^2m alpha`
`=> theta = n pi + (-1)^(n) alpha , n in I`
(ii) `cos theta = cos alpha=> cos alpha - cos theta =0 => 2 sin ((alpha + theta)/2) sin ((theta - alpha)/2) =0`
`=> sin (theta + alpha)/2 =0 ` or `sin (theta -alpha)/2 =0 , (theta+ alpha)/2 = n pi` or `(theta - alpha) /2 = n pi`
`=> theta = 2 n pi -alpha ` or `theta = 2 n pi+ alpha n in I`
`=> theta = 2 n pi pm alpha`
(iii) `tan theta = tan alpha => sin theta/cos theta = sin alpha /cos alpha => sin theta cos theta- cos theta * sin alpha =0`
`=> sin (theta -alpha) =0 => theta - alpha = n pi`
`=> theta = n pi+ alpha ` , where `n in I`
(iv) `sin^2 theta = sin^2 alpha `
`sin^2 theta - sin^2 alpha = sin (theta+ alpha) sin (theta - alpha) =0`
`sin (theta+ alpha )=0` or `sin (theta -alpha)=0`
`theta+ alpha = n pi ` or `theta - alpha = n pi, n in I`
`theta = n pi pm alpha , n in I`
(v) `cos ^2 theta = cos^2 alpha => - sin^2 theta -1 -sin^2 alpha => sin^2 theta - sin^2 alpha`
`theta = n pi pm alpha , n in I`
(vi) `tan^2 theta = tan^2 alpha => tan theta = pm tan alpha = tan(pm alpha)`
`=> theta = n pi pm alpha ` , where `n in I`
(ii) If `cos theta = cos alpha => theta = 2 n pi pm alpha`, where ` alpha in [0, pi], n on I`
(iii) If `tan theta = tan alpha => theta = n pi + alpha ` , where `alpha in (-pi/2, pi/2), n in I`
(iv) If `sin^2 theta = sin^2 alpha => theta = n pi pm alpha `
(v) lf `cos^2 theta = cos^2 alpha => n pi pm alpha`
(vi) lf `tan^2 theta = tan^2 alpha => theta = n pi pm alpha`
Note : `alpha` is called the principal angle.
Proof :
(i) `sin theta = sin alpha => sin theta - sin alpha =0 => 2 cos ((theta+alpha)/2) sin ((theta - alpha)/2) =0`
`=> cos ((theta+alpha)/2) =0 ` or `sin ( (theta - alpha)/2)=0`
`=> (theta+ alpha)/2 =(2m+1) pi/2 ` or `(theta - alpha)/2 = m pi`, where `m in I`
`=> theta = (2m +1) pi - alpha ` or `theta = 2m pi + alpha` , where `m in I`
`=> theta = (2m+1) pi +(-1)^(2m+1) alpha ` or `theta =2 m pi+ (-1)^2m alpha`
`=> theta = n pi + (-1)^(n) alpha , n in I`
(ii) `cos theta = cos alpha=> cos alpha - cos theta =0 => 2 sin ((alpha + theta)/2) sin ((theta - alpha)/2) =0`
`=> sin (theta + alpha)/2 =0 ` or `sin (theta -alpha)/2 =0 , (theta+ alpha)/2 = n pi` or `(theta - alpha) /2 = n pi`
`=> theta = 2 n pi -alpha ` or `theta = 2 n pi+ alpha n in I`
`=> theta = 2 n pi pm alpha`
(iii) `tan theta = tan alpha => sin theta/cos theta = sin alpha /cos alpha => sin theta cos theta- cos theta * sin alpha =0`
`=> sin (theta -alpha) =0 => theta - alpha = n pi`
`=> theta = n pi+ alpha ` , where `n in I`
(iv) `sin^2 theta = sin^2 alpha `
`sin^2 theta - sin^2 alpha = sin (theta+ alpha) sin (theta - alpha) =0`
`sin (theta+ alpha )=0` or `sin (theta -alpha)=0`
`theta+ alpha = n pi ` or `theta - alpha = n pi, n in I`
`theta = n pi pm alpha , n in I`
(v) `cos ^2 theta = cos^2 alpha => - sin^2 theta -1 -sin^2 alpha => sin^2 theta - sin^2 alpha`
`theta = n pi pm alpha , n in I`
(vi) `tan^2 theta = tan^2 alpha => tan theta = pm tan alpha = tan(pm alpha)`
`=> theta = n pi pm alpha ` , where `n in I`