Answer:
The identity is proved below.
Step-by-step explanation:
The tangent of an angle [tex]\theta[/tex] is given by:
[tex]\tan{\theta} = \frac{\sin{\theta}}{\cos{\theta}}[/tex]
In this question, we are given the following trigonometric identity:
[tex]\frac{\sin{\theta}}{\sin{\theta}+\cos{\theta}} = \frac{\tan{\theta}}{1+\tan{\theta}}[/tex]
Applying the tangent
[tex]\frac{\sin{\theta}}{\sin{\theta}+\cos{\theta}} = \frac{\frac{\sin{\theta}}{\cos{\theta}}}{1+\frac{\sin{\theta}}{\cos{\theta}}}[/tex]
Now, we apply the least common multiple on the denominator. So
[tex]\frac{\sin{\theta}}{\sin{\theta}+\cos{\theta}} = \frac{\frac{\sin{\theta}}{\cos{\theta}}}{\frac{\cos{\theta}+\sin{\theta}}{\cos{\theta}}}[/tex]
SImplifying the cosine:
[tex]\frac{\sin{\theta}}{\sin{\theta}+\cos{\theta}} = \frac{\sin{\theta}}{\sin{\theta}+\cos{\theta}}[/tex]
Which means that the identity is proved.
