Trigonometry Chapter 3~ Properties of Trigonometric Functions

Example 1 (3.1) Prove that 1+tan²x = sec²x

1+ sin²x/ cos²x= sec²x , cos²x/cos²x + sin²x/sin²x= sec²x , cos²x +sin²x/ cos²x = sec² x , 1/cos²x= sec²x . Sec²x= sec²x

Example 1 (3.2) Prove 1+t²/1-t² - t/1-t = 1/1+t

1+t²/(1+t)(1-t) - t/1-t =1/1+t => 1+t²-t-t²/(1+t)(1-t)=1/1+t =>1-t/(1+t)(1-t) = 1/1+t=> 1/1+t=1/1+t

Example 3 (3.1) Express 1 - sin x cos x tan x in terms of cos x

1-sin x *cos x * sin x/ cos x = 1-sin²x = cos²x

Strategies for Proving Identities

1. If one side of a trig. identity is a sum or difference of two rational expressions, try combing over a common denominator. 2. Try converting one side of the identity into an expression in terms of just sin x and/or cos x. 3. If one side of the identity is given in factored form, try multiplying the parts of the factored expression. 4. If one side of the identity can be factored,try factoring that side. Keep in mind:a²-b²= (a+b)(a-b) or a²+2ab+b²= (a+b)² or a²-2ab+b²= (a-b)² . 5. Work with the more complicated side of the equation first. 6. The presence of squared functions (i:e.:sin²x) should suggest the use of one of the Pythagorean Identities. 7. Remember that you can use the alternative forms of certain identities(because of sin²x+cos²x=1, you can use sin²x=1-cos²x). 8. At last resort,try simplifying both sides of the equation to the same expression.

Example 2 (3.2) Prove sec x-cos x= sin x*tan x

1/cos x- cos x/1= sin x*tan x, 1/cos x- cos²x/cos x= sin x *tan x, 1-cos²x/cos x= sin x *tan x, sin²x/cos x = sin*sin x/cos x => sin²x/cos x = sin²x/cos x

Example 2 (3.1) Express csc²x + cot²x in terms of sin x

1/sin²x + cos²x/ sin²x , 1+cos²x/ sin²x => 1+ (1-sin²x)/sin²x => 2-sin²x/ sin²x

Difference Identity for cos

Cos(a-b)= cos a*cos b+sin a*sin b

Difference Identity for sin

Sin (a-b)= sin a*cos b - cos a*sin b

Trigonometric identities

Tan x= sin x/cos x and cot x= cos x/sin x

Double Angle Identity cos

cos (2x) = cos²x- sin²x or cos(2x)=2cos²x-1 or cos(2x)= 1- 2sin²x

Sum identity for cos

cos (a+b)= cos a*cos b- sin a *sin b

Half angle identity cos

cos (x/2) =±√1+cos x/2

Double Angle Identity sin

sin (2x) = 2 sin x cos x

Sum Identity for sin

sin (a+b)=sin a * cos b+cos a* sin b

Half angle identity sin

sin (x/2) =±√1-cos x/2

Reciprocal Identities

sin x = 1/csc x , csc x = 1/sin x , cos x = 1/sec x , sec x = 1/cos x , tan x = 1/cot x , cot x = 1/tan x

Example 3 (3.2) sin x/csc x + cos x/sec x= sec²x-tan²x

sin x/1+sin x +cos x/1/cos x = sec²x-tan²x => sin²x+cos²x= 1+tan²x-tan²x , 1=1

Pythagorean Identities

sin²x + cos²x = 1 , 1+tan²x= sec²x , 1+cot²x=csc²x , (sin²x= 1-cos²x) , (tan²x= sec²x-1) , (cot²x= csc²x-1) , (cos²x= 1-sin²x) , (sec²x- tan²x=1) , (csc²x - cot²x =1)

Double Angle Identity for tan

tan (2x)= 2 tan x/ 1-tan²x

Sum identity for tan

tan (a+b) = tan a+tan b/1-tan a*tan b

Difference identity for tan

tan (a-b)= tan a-tan b/ 1+tan a *tan b

Half Angle Identity for tan

tan (x/2) = sin x/ 1+cos x or tan (x/2)= 1-cos x/sin x or tan(x/2)= ±√1-cos x/1+cos x