http://www.ck12.org Chapter 6. Analytic Trigonometry
sin^2 x
cos^2 x+cos^2 x
cos^2 x=1
cos^2 x
tan^2 x+ 1 =sec^2 x(sec^2 x)( 1 −sin^2 x)−(sinx
cscx+cosx
secx)
=sec^2 x·cos^2 x−(sin^2 x+cos^2 x)
= 1 − 1
= 0- Note that initially, the expression is not the same as the Pythagorean identity.
(cost−sint)^2 +(cost+sint)^2
=cos^2 t−2 costsint+sin^2 t+cos^2 t+2 costsint+sin^2 t
= 1 −2 costsint+ 1 +2 costsint
= 2Practice
Prove each of the following:
1.( 1 −cos^2 x)( 1 +cot^2 x) = 1
cosx( 1 −sin^2 x) =cos^3 x
sin^2 x= ( 1 −cosx)( 1 +cosx)
sinx=sin^2 xcsc+cosx^2 x
sin^4 x−cos^4 x=sin^2 x−cos^2 x
sin^2 xcos^3 x= (sin^2 x−sin^4 x)(cosx)
Simplify each expression as much as possible.
tan^3 xcsc^3 x
8.cscsec^2 x 2 −x^1
9.^11 −+sinsin^2 xx
√
1 −cos^2 x
11.sin^2 cosx− 2 sinx^4 x
12.( 1 +tan^2 x)(sec^2 x)
13.sin^2 x+tansec^2 xx+cos^2 x
14.^1 +csctan (^22) xx
15.^1 −cossinx^2 x