Home > Trigonometry > Fundamental Identities

How do you simplify #(sin theta csc theta)/cot theta#?

Answer 1

Madelyn Adams

#color(green)(tan theta#

From the above trigonometric identities,

# csc theta = (1 / sin theta), cot theta = 1/ tan theta = cos theta / sin theta#

Substituting the above in the given sum,

#(cancel sin theta * (1/ cancel sin theta) ) / (cos theta / sin theta)#

#=> sin theta / cos theta = color(green)(tan theta#

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

Answer 2

Caroline Adams

#(sin theta csc theta)/cot theta = tan theta#.

Firstly, the cosecant function is defined as the inverse sine function: #csc varphi = 1/sin varphi# and the cotangent function as the inverse tangent function: #cot varphi = 1/tanvarphi = 1/(sinvarphi/cosvarphi) = cos varphi/sin varphi#.

Substituting it into the equation, we have

#(sin theta * 1/color(red)(sin theta))/color(red)(cos theta/sintheta)#

We know that the product of a number and its inverse is always #1# :

#color(red)1/color(red)(costheta/sintheta) = sin theta/cos theta = tan theta#.

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.