Are the following equations or identities?

Question

1) $tan x = \frac{sin x}{cos x}$ $tan x=sin x/cos x$

2) $sec x = \frac{sin x}{cos x}$ $secx=sinx/cosx$ .

3) $csc x = \frac{1}{sin x}$ $cscx = 1/sinx$

4) $sec x = \frac{1}{\frac{1}{sin x}}$ $secx = 1/(1/sinx)$

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Answer 1 · 2018-01-29T23:06:54

1) identity
2) equation.
3) identity
4) equation.

1) By the definition, (basic trigonometry) we have:

$tan x = \frac{sin x}{cos x}$ $tan x=sin x/cos x$

so the first statement is an identity

2) By the definition, (basic trigonometry) we have:

$sec x = \frac{1}{cos x} \neq \frac{sin x}{cos x}$ $secx=1/cosx != sinx/cosx$

so the second statement is not an identity, and is therefore an equation.

3) By the definition, (basic trigonometry) we have

$csc x = \frac{1}{sin x}$ $cscx = 1/sinx$

so the third statement is an identity

4) By the definition, (basic trigonometry) we have

$sec x = \frac{1}{cos x} \neq \frac{1}{\frac{1}{sin x}}$ $secx = 1/cosx != 1/(1/sinx)$

so the fourth statement is not an identity, and is therefore an equation.