Prove that the following identity is true. sin2 tan x x sin x cos x = sin x - cos x cos x 2 cos x We begin on the right side of the equation by factoring the numerator and denominator. We can then use a Pythagorean Identity in the denominator and reduce. We can then use the ratio identity, and then factor the denominator. Finally we simplify by reducing the common factor. sin x(sin x + cos x) sin2 xsin x cos x Cos X 1 - cos X 2 cos3 x cos x) sin x(sin x = sin2 xcos2 x COS X sin x cos x sin x sin2 x - COS X sin x cos x = tan x sin2 x sin x cos x = tan x. (sin x - cos x) tan x sin x - cos x
Prove that the following identity is true. sin2 tan x x sin x cos x = sin x - cos x cos x 2 cos x We begin on the right side of the equation by factoring the numerator and denominator. We can then use a Pythagorean Identity in the denominator and reduce. We can then use the ratio identity, and then factor the denominator. Finally we simplify by reducing the common factor. sin x(sin x + cos x) sin2 xsin x cos x Cos X 1 - cos X 2 cos3 x cos x) sin x(sin x = sin2 xcos2 x COS X sin x cos x sin x sin2 x - COS X sin x cos x = tan x sin2 x sin x cos x = tan x. (sin x - cos x) tan x sin x - cos x
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter6: The Trigonometric Functions
Section6.4: Values Of The Trigonometric Functions
Problem 42E
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![Prove that the following identity is true.
sin2
tan x
x sin x cos x
=
sin x - cos x
cos x 2 cos x
We begin on the right side of the equation by factoring the numerator and denominator. We can then use a Pythagorean Identity in the denominator and reduce. We can then use the
ratio identity, and then factor the denominator. Finally we simplify by reducing the common factor.
sin x(sin x + cos x)
sin2 xsin x cos x
Cos X 1 -
cos X 2 cos3 x
cos x)
sin x(sin x
=
sin2 xcos2 x
COS X
sin x
cos x
sin x
sin2 x -
COS X
sin x cos x
= tan x
sin2 x
sin x cos x
= tan x.
(sin x - cos x)
tan x
sin x - cos x](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F07855119-4567-46b4-87d0-bc5ee906db91%2F5224222f-5b61-4513-98fd-6b6800b8ef44%2Ffh4lcl.png&w=3840&q=75)
Transcribed Image Text:Prove that the following identity is true.
sin2
tan x
x sin x cos x
=
sin x - cos x
cos x 2 cos x
We begin on the right side of the equation by factoring the numerator and denominator. We can then use a Pythagorean Identity in the denominator and reduce. We can then use the
ratio identity, and then factor the denominator. Finally we simplify by reducing the common factor.
sin x(sin x + cos x)
sin2 xsin x cos x
Cos X 1 -
cos X 2 cos3 x
cos x)
sin x(sin x
=
sin2 xcos2 x
COS X
sin x
cos x
sin x
sin2 x -
COS X
sin x cos x
= tan x
sin2 x
sin x cos x
= tan x.
(sin x - cos x)
tan x
sin x - cos x
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