Cheatography
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A quick reference guide for solving trigonometric proofs and some basic trig identities to use.
Solving Trigonometric Proofs
Method 1 
Pick one side of the equation (usually the most complicated side) and work with that side until it is equal to the other side 
Method 2 
Work with both sides simultaneously until they are both equal to the same expression. 
Helpful Techniques
Rewrite Rewrite the expression in terms of sine and cosine only

Multiply by One Multiply the numerator and denominator of a rational expression by a carefully chosen "one"

Combine fractions Write sums and differences of rational expressions as a single fraction

Factor Factor trigonometric expressions, using "usubstitution" as needed

Pythagorean Theorem Look for combinations or portions of Pythagorean Identities. Remember that you can multiply, divide, add or subtract the identity to get a new version.

Goal Always keep the goal in mind. As you manipulate one side of the equation, keep the other side in mind. Look for commonalities

Verifying Trigonometric Identities


Pythagorean Identities
sin^{2}x + cos^{2}x = 1
sin^{2}x = 1  cos^{2}x
cos^{2}x = 1  sin^{2}x
1 + cot^{2}x = csc^{2}x
cot^{2}x = csc^{2}x  1
tan^{2}x + 1 = sec^{2}x
tan^{2}x = sec^{2}x  1 
Divide original Pythagorean identity by sin^{2}x or cos^{2}x to get other identities, subtract to get even more.
Basic Trigonometric Functions
sin (x) = sin (x) 
tan (x) = sin (x) / cos (x) 
csc (x) = 1/sin (x) 
cos (x) = cos (x) 
cot (x) = cos (x) / sin (x) 
sec (x) = 1/cos (x) 
tan (x) = tan (x) 

cot (x) = 1/tan (x) 
Complementary Angle Identities
sin (pi/2  x) = cos (x) 
cos (pi/2  x) = sin (x) 
tan (pi/2  x) = cot (x) 
Sum and Difference Identities
sin (a +/ b) = 
sin (a) cos (b) +/ cos (a) sin (b) 
cos (a +/ b) = 
cos (a) cos (b) /+ sin (a) sin (b) 
Use sin (a +/ b) / cos (a +/ b) to find tan (a +/ b)
Double Angle Identities
sin (2x) 
cos (2x) 
2 sin (x) cos (x) 
cos^{2} (x)  sin^{2} (x) 

1  2 sin^{2} (x) 

2 cos^{2} (x)  1 

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duckduckBane, 15:11 20 Dec 21
Good stuff
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