Verify that sec(θ)/csc(θ)-cot(θ) - sec(θ)/csc(θ)-cot(θ) = 2csc(θ) is an identity.
please help! thank you!
sec(θ)/(csc(θ)-cot(θ)) - sec(θ)/(csc(θ)+cot(θ))
becomes
sec(θ)(csc(θ)+cot(θ)) - sec(θ)(csc(θ)-cot(θ))
----------------------------------------------------------
(csc(θ)-cot(θ))(csc(θ)+cot(θ))
Now expand the numerator and note that the denominator is just 1.
To verify the given statement that sec(θ)/csc(θ) - cot(θ) - sec(θ)/csc(θ) - cot(θ) = 2csc(θ) is an identity, we need to simplify both sides of the equation and show that they are equal.
Let's start by simplifying the left side of the equation step by step:
sec(θ)/csc(θ) - cot(θ) - sec(θ)/csc(θ) - cot(θ)
Now, to simplify sec(θ)/csc(θ), we can use the reciprocal identities:
sec(θ)/csc(θ) = 1/cos(θ) / 1/sin(θ)
= sin(θ) / cos(θ) * sin(θ) / 1
= sin^2(θ) / cos(θ)
Therefore, the expression becomes:
(sin^2(θ) / cos(θ)) - cot(θ) - (sin^2(θ) / cos(θ)) - cot(θ)
Now, let's simplify cot(θ) as well:
cot(θ) = 1/tan(θ)
= cos(θ) / sin(θ)
Substituting this back into the expression:
(sin^2(θ) / cos(θ)) - (cos(θ) / sin(θ)) - (sin^2(θ) / cos(θ)) - (cos(θ) / sin(θ))
Next, we can combine like terms:
(sin^2(θ) - sin^2(θ)) / cos(θ) - (cos(θ) + cos(θ)) / sin(θ)
The sin^2(θ) terms cancel out:
0 / cos(θ) - (2cos(θ)) / sin(θ)
0 - (2cos(θ)) / sin(θ)
= -2cos(θ) / sin(θ)
Finally, we simplify the right side of the equation:
2csc(θ)
Using the reciprocal identities, csc(θ) = 1/sin(θ):
2 * (1/sin(θ))
= 2/sin(θ)
Now we can see that the left side of the equation, -2cos(θ)/sin(θ), is equal to the right side of the equation, 2/sin(θ).
Thus, we have verified that sec(θ)/csc(θ) - cot(θ) - sec(θ)/csc(θ) - cot(θ) = 2csc(θ) is an identity.