How to Simplify Mathematical Expressions Using Even/Odd Identities

If you are a student of Mathematics or Further Mathematics, then you cannot help but come across trigonometric identities – Sine, Cosine and Tangent. So what should you do when they are presented in mathematical problems? Just hold on, dear student, live and learn.

Because sine, cosine, and tangent are functions (trig functions), they can be defined as even or odd functions as well. Sine and tangent are both odd functions, and cosine is an even function. In other words,

• sin(–x) = –sin x

• cos(–x) = cos x

• tan(–x) = –tan x

These identities will all make appearances in problems that ask you to simplify an expression, prove an identity, or solve an equation. So what’s the big red flag? The fact that the variable inside the trig function is negative. When tan(–x), for example, appears somewhere in an expression, it should usually be changed to –tan x.

The following steps show you how to simplify [fusion_builder_container hundred_percent=”yes” overflow=”visible”][fusion_builder_row][fusion_builder_column type=”1_1″ background_position=”left top” background_color=”” border_size=”” border_color=”” border_style=”solid” spacing=”yes” background_image=”” background_repeat=”no-repeat” padding=”” margin_top=”0px” margin_bottom=”0px” class=”” id=”” animation_type=”” animation_speed=”0.3″ animation_direction=”left” hide_on_mobile=”no” center_content=”no” min_height=”none”][1 + sin(–x)][1 – sin(–x)]:

1. Get rid of all the –x values inside the trig functions.

   You see two sin(–x) functions, so you replace them both with –sin x to get: [1 + (–sin x)][1 – (–sin x)].

2. Simplify the new expression.

   First adjust the two negative signs within the parentheses to get (1 – sin x)(1 + sin x), and then open the brackets to get 1 – sin² x.

3. Look for any combination of terms that could give you a Pythagorean identity.

   Whenever you see a function squared, you should think of the Pythagorean identities. The three Pythagorean identities are

   

   Looking at the identities, you see that 1 – sin² x is the same as cos² x.

4.  Now the expression is fully simplified as cos² x.

Note: You can use of any of these three equations to find the others. Just remember that

1/tan² x = cot² x and 1/tan x = cot x;  1/cos² x = sec² x and 1/cos x = sec x;  1/sin² x = cosec² x and 1/sin x = cosec x

This article was adapted from dummies.com

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