Home > Trigonometry > Fundamental Identities

How do you simplify #Tan(x) Csc(x)/sec(x)#?

Answer 1

Samantha Abate

Simplify: #tan x(csc x)/(sec x)#

Ans: 1

#tan x(1/sin x)/(1/cos x) = # #(sin x/cos x)(1/sin x)(cos x/1)# = 1

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Brooklyn Anderson

To simplify (\tan(x) \csc(x) / \sec(x)), use trigonometric identities:

(\csc(x) = \frac{1}{\sin(x)})
(\sec(x) = \frac{1}{\cos(x)})

Substitute these identities into the expression:

[\frac{\tan(x) \csc(x)}{\sec(x)} = \frac{\tan(x) \frac{1}{\sin(x)}}{\frac{1}{\cos(x)}}]

Now, multiply the numerator and denominator by (\sin(x)) to rationalize:

[\frac{\tan(x) \frac{1}{\sin(x)}}{\frac{1}{\cos(x)}} \times \frac{\sin(x)}{\sin(x)} = \frac{\tan(x)}{\frac{\sin(x)}{\cos(x)}}]

Using the identity (\tan(x) = \frac{\sin(x)}{\cos(x)}):

[\frac{\tan(x)}{\frac{\sin(x)}{\cos(x)}} = \frac{\tan(x)}{\tan(x)} = \boxed{1}]

Therefore, (\tan(x) \csc(x) / \sec(x)) simplifies to (1).

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.