How do you simplify the expression $(sintheta+tantheta)/(1+sectheta)$ ?

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Answer 1 · 2016-09-09T14:30:36

$sin(theta)$

We have: $(sin(theta) + tan(theta)) / (1 + sec(theta))$

Let's apply two standard trigonometric identities; $tan(theta) = (sin(theta)) / (cos(theta))$ and $sec(theta) = (1) / (cos(theta))$ :

$= (sin(theta) + (sin(theta)) / (cos(theta))) / (1 + (1) / (cos(theta)))$

$= ((sin(theta)cos(theta) + sin(theta)) / (cos(theta))) / ((cos(theta) + 1) / (cos(theta)))$

$= (sin(theta)cos(theta) + sin(theta)) / (cos(theta) + 1)$

Let's apply the Pythagorean identity $cos^(2)(theta) + sin^(2)(theta) = 1$ :

$= (sin(theta) (cos(theta) + 1)) / (cos(theta) + cos^(2)(theta) + sin^(2)(theta))$

We can rearrange the Pythagorean identity to get:

$=> sin^(2)(theta) = 1 - cos^(2)(theta)$

Let's apply this to get:

$= (sin(theta) (cos(theta) + 1)) / (cos(theta) + cos^(2)(theta) + 1 - cos^(2)(theta))$

$= (sin(theta) (cos(theta) + 1)) / (cos(theta) + 1)$

$= sin(theta)$

How do you simplify the expression (sintheta+tantheta)/(1+sectheta)(sintheta+tantheta)/(1+sectheta)?