Simplifying Trigonometric Expressions

The command Simplify combines and simplifies trigonometric expressions, as in the following examples.

$\blacktriangleright$ Simplify

cos2x + ${\frac{{1}}{{4}}}$sin22x - sin2x cos2x + 2 sin2x =   - cos2x + 2

$\left(\vphantom{ \cos 3t+3\cos t}\right.$cos 3t + 3 cos t$\left.\vphantom{ \cos 3t+3\cos t}\right)$sec t =  4 cos2t

sin 3a + 4 sin3a =  3 sin a

tan${\frac{{\theta }}{{2}}}$sin${\frac{{\theta }}{{2}}}$ + cos${\frac{{\theta }}{{2}}}$ -2 cscθsin${\frac{{\theta }}{{2}}}$ =   0

You may need to apply repeated operations to get the result you want. The order in which you apply the operations is not necessarily critical. You achieve the first of the following examples by applying Simplify followed by Expand. For the second example, apply Expand followed by Simplify.

$\blacktriangleright$ Simplify, Expand

$\left(\vphantom{ \sec t}\right.$sec t$\left.\vphantom{ \sec t}\right)$$\left(\vphantom{ 1+\cos 2t}\right.$1 + cos 2t$\left.\vphantom{ 1+\cos 2t}\right)$ =  ${\dfrac{{1}}{{\cos t}}}$$\left(\vphantom{ 1+\cos 2t}\right.$1 + cos 2t$\left.\vphantom{ 1+\cos 2t}\right)$ =  2 cos t

$\blacktriangleright$ Expand, Simplify

$\left(\vphantom{ \sec t}\right.$sec t$\left.\vphantom{ \sec t}\right)$$\left(\vphantom{ 1+\cos 2t}\right.$1 + cos 2t$\left.\vphantom{ 1+\cos 2t}\right)$ =  2 sec t cos2t =  2 cos t 4pt