Verify the trigonometric identity
'squareroot' 1+sin (theta) / 1 - sin (theta) = 1+sin (theta) / | cos(theta) |
+23251 Verify the identity: sqrt( [1 + sin(x)] / [1 - sin(x)] ) = [1 + sin(x)] / | cos(x) |
Start with the left-hand side:
sqrt( [1 + sin(x)] / [1 - sin(x)] )
Multiply both the numerator and the denominator of the interior of this square root by the conjugate of the denominator
the conjugate of 1 - sin(x) is 1 + sin(x)
= sqrt( [ ( 1 + sin(x) )( 1 + sin(x) ) ] / [ ( 1 - sin(x) ) / ( 1 + sin(x) ) ]
= sqrt( [ ( 1 + sin(x) )2 ] / [ 1 - sin2(x) ] )
= sqrt( [ ( 1 + sin(x) )2 ] / [ cos2(x) ] )
Divide this into two parts, the numerator and the denominator:
= sqrt( ( 1 + sin(x) )2 ) / sqrt ( cos2(x) )
Simplify the square roots:
= ( 1 + sin(x) ) / | cos(x) |
