Which of the following is not an identity?
A. $\cos ^2 x \csc x-\csc x=-\sin x$
B. $\sin x(\cot x+\tan x)=\sec x$
C. $\csc ^2 x+\sec ^2 x=1$
D. $\cos ^2 x-\sin ^2 x=1-2 \sin ^2 x$
Answer (1)
Verify each equation separately.
A. cos 2 x csc x − csc x = − sin x is an identity.
B. sin x ( cot x + tan x ) = sec x is an identity.
C. csc 2 x + sec 2 x = 1 is not an identity.
D. cos 2 x − sin 2 x = 1 − 2 sin 2 x is an identity. C
Explanation
Understanding the Problem We are given four equations and asked to identify the one that is not an identity. An identity is an equation that is true for all values of the variable. We need to verify each equation to see if it holds true for all x.
Solution Plan We will verify each equation separately to determine which one is not an identity.
Verifying Option A A. cos 2 x csc x − csc x = − sin x . We can rewrite this in terms of sin x and cos x to verify if it's an identity.
Calculations for Option A cos 2 x csc x − csc x = s i n x c o s 2 x − s i n x 1 = s i n x c o s 2 x − 1 . Since cos 2 x + sin 2 x = 1 , we have cos 2 x − 1 = − sin 2 x . Therefore, s i n x c o s 2 x − 1 = s i n x − s i n 2 x = − sin x . So, option A is an identity.
Verifying Option B B. sin x ( cot x + tan x ) = sec x . We can rewrite this in terms of sin x and cos x to verify if it's an identity.
Calculations for Option B sin x ( cot x + tan x ) = sin x ( s i n x c o s x + c o s x s i n x ) = sin x ( s i n x c o s x c o s 2 x + s i n 2 x ) = sin x ( s i n x c o s x 1 ) = c o s x 1 = sec x . So, option B is an identity.
Verifying Option C C. csc 2 x + sec 2 x = 1 . We can rewrite this in terms of sin x and cos x to verify if it's an identity.
Calculations for Option C csc 2 x + sec 2 x = s i n 2 x 1 + c o s 2 x 1 = s i n 2 x c o s 2 x c o s 2 x + s i n 2 x = s i n 2 x c o s 2 x 1 . This is not equal to 1 for all x , so option C is not an identity. For example, if x = 4 π , then sin 2 x = cos 2 x = 2 1 , so s i n 2 x c o s 2 x 1 = 4 1 1 = 4 = 1 .
Verifying Option D D. cos 2 x − sin 2 x = 1 − 2 sin 2 x . We can use the identity cos 2 x + sin 2 x = 1 to verify if it's an identity.
Calculations for Option D cos 2 x − sin 2 x = ( 1 − sin 2 x ) − sin 2 x = 1 − 2 sin 2 x . So, option D is an identity.
Conclusion Therefore, option C is not an identity.
Examples
Trigonometric identities are fundamental in various fields such as physics, engineering, and computer graphics. For example, in physics, they are used to simplify complex equations describing wave phenomena, such as light and sound. In engineering, they are crucial for analyzing and designing electrical circuits and mechanical systems. In computer graphics, trigonometric identities help in rendering 3D objects and creating realistic animations by manipulating angles and positions.
