Lecture Notes Trigonometric Identities page Sample Problems Prove each of the following identities.. tan x x + sec x 2. tan x + tan x x 3. x x 3 x 4. 5. + + + x 6. 2 sec + x 2 tan x csc x tan x + cot x 0. 2 tan2 x tan 2 x +. tan 2 csc 2 tan 2 2. sec x + tan x x 3. csc cot tan 4. 4 x 4 x 2 5. ( x ) 2 + ( x + ) 2 2 7. 4 x 4 x 6. + 4 x + 3 3 + x x 8. tan 2 x tan 2 x + 2 x 7. x tan x sec x 9. x + x 8. tan 2 x + + tan x sec x + x c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 203
Lecture Notes Trigonometric Identities page 2 Practice Problems Prove each of the following identities.. tan x + + x. cot x cot x + tan x + tan x 2. tan 2 x + sec 2 x 2. ( x + ) (tan x + cot x) sec x + csc x 3. x + x 2 tan x sec x 3. 3 x + 3 x x + x 4. tan x + cot x sec x csc x 5. + tan 2 x tan 2 x 4. 5. + 3 x csc x + x x x + x 4 tan x sec x 6. tan 2 x tan 2 x 6. csc 4 x cot 4 x csc 2 x + cot 2 x 7. x + x 2 csc x 7. + 3 + 2 2 + 8. sec x sec x + + 8. tan x + tan y cot x + cot y tan x tan y 9. + cot 2 x csc 2 x 0. csc 2 x csc 2 x 9. + tan x tan x + x x 20. ( x tan x) ( cot x) ( x ) ( ) c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 203
Lecture Notes Trigonometric Identities page 3 Sample Problems - Solutions. tan x x + sec x We will only use the fact that + for all values of x. LHS tan x x + x x + 2 x + 2 x + 2 x 2 x + RHS 2. tan x + tan x x We will only use the fact that + for all values of x. 3. x x 3 x LHS + tan x tan x x + x 2 x + x We will only use the fact that + for all values of x. x RHS LHS x x x x RHS 4. + + + 2 sec We will only use the fact that + for all values of x. 5. LHS x + + + 2 ( + )2 + ( + ) ( + ) 2 + ( + ) 2 ( + ) 2 + + 2 + 2 2 + 2 + + 2 ( + ) ( + ) 2 ( + ) ( + ) 2 2 2 sec RHS + x 2 tan x 2 + 2 ( + ) We will start with the left-hand side. First we bring the fractions to the common denominator. Recall that + for all values of x. LHS x ( + x) ( x) ( x) ( + x) 2 x + x ( + x) ( x) ( + x) 2 tan x RHS ( x) ( x) ( + x) + x + x 2 x c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 203
Lecture Notes Trigonometric Identities page 4 6. csc x tan x + cot x We will start with the right-hand side. We will re-write everything in terms of x and and simplify. We will again run into the Pythagorean identity, +. RHS csc x tan x + cot x x x x x + x 2 x x x + LHS 2 x x x + x x x 7. 8. 4 x 4 x We can factor the numerator via the di erence of squares theorem. LHS 4 x 4 2 x 2 x () 2 + RHS tan 2 x tan 2 x + 2 x 2 x + LHS tan 2 x tan 2 x + + 2 x 2 x + + 2 x 2 x + 2 x RHS 9. x + x LHS x x x ( + x) + x RHS + x + x ( x) ( + x) ( + x) ( + x) c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 203
Lecture Notes Trigonometric Identities page 5 0. 2 tan2 x tan 2 x + RHS tan2 x tan 2 x + 2 x + 2 x + 2 x + 2 x + 2 x + 2 x 2 LHS. tan 2 csc 2 tan 2 RHS csc 2 tan 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 tan 2 LHS 2. sec x + tan x x 3. csc RHS cot tan x x x + x + x ( + x) ( x) ( + x) ( + x) ( + x) 2 + x x + x LHS We will start with the left-hand side. We will re-write everything in terms of and and simplify. We will again run into the Pythagorean identity, + for all angles x. LHS csc 2 2 cot tan 4. 4 x 4 x 2 2 + 2 2 2 2 2 RHS 2 2 2 LHS 4 x 4 x 2 2 + 2 RHS c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 203
Lecture Notes Trigonometric Identities page 6 5. ( x ) 2 + ( x + ) 2 2 LHS ( x ) 2 + ( x + ) 2 + 2 x + + + 2 x 2 + 2 2 + 2 2 RHS 6. + 4 x + 3 3 + x x LHS 2 x + 4 x + 3 ( x + ) ( x + 3) ( x + ) ( x + 3) ( + x) ( x) x + 3 x RHS 7. x LHS tan x sec x x tan x 2 x + x ( x) x x 2 x x ( x) ( x) x ( x) RHS 2 x x + ( x) 8. tan 2 x + + tan x sec x + x LHS tan 2 x + + tan x sec x 2 x + + x 2 x + 2 x + x 2 x + + x + x RHS For more documents like this, visit our page at http://www.teaching.martahidegkuti.com and click on Lecture Notes. E-mail questions or comments to mhidegkuti@ccc.edu. c copyright Hidegkuti, Powell, 2009 Last revised: May 8, 203