Determine a formula for sec(2è) using both sec(è) and csc(è).
Determine a formula for csc(2è) using both sec(è) and csc(è).
To determine a formula for sec(2θ) using both sec(θ) and csc(θ), we can use the double-angle identity for secant, which states:
sec(2θ) = 1 / cos(2θ)
Now let's use the double-angle identity for cosine:
cos(2θ) = cos²(θ) - sin²(θ)
First, we need to express the formula in terms of sec(θ) and csc(θ), so let's rewrite it using the reciprocal identities:
cos(2θ) = 1 - sin²(θ) / cos²(θ)
Now, recall that secant is the reciprocal of cosine:
sec(θ) = 1 / cos(θ)
Hence, we have:
cos(2θ) = 1 - sin²(θ) / (1 / sec²(θ))
Replacing the terms with sec(θ) and csc(θ):
cos(2θ) = 1 - (1 / csc²(θ)) / (1 / sec²(θ))
Simplifying the equation:
cos(2θ) = 1 - sec²(θ) / csc²(θ)
Finally, using the reciprocal identity for tangent:
sec(θ) = 1 / cos(θ) = cos(θ) / (cos²(θ) - sin²(θ))
And the reciprocal identity for cotangent:
csc(θ) = 1 / sin(θ) = cos(θ) / (cos²(θ) - sin²(θ))
We can rewrite the equation as:
cos(2θ) = 1 - (sec²(θ) / csc²(θ))
= 1 - (cos²(θ) / sin²(θ)) / (cos²(θ) / sin²(θ))
= 1 - (cos²(θ) / sin²(θ)) * (sin²(θ) / cos²(θ))
= 1 - 1
= 0
Therefore, the formula for sec(2θ) in terms of sec(θ) and csc(θ) is 0.
To determine a formula for csc(2θ) using both sec(θ) and csc(θ), we can use the double-angle identity for cosecant, which states:
csc(2θ) = 2 / (2 * sin(θ) * cos(θ))
Now let's rewrite the formula using the reciprocal identities for sine and cosine:
csc(2θ) = 1 / (sin(θ) * cos(θ))
Next, let's express the formula in terms of sec(θ) and csc(θ):
csc(2θ) = 1 / (1 / csc(θ) * 1 / sec(θ))
Using the reciprocal identity for sine:
csc(θ) = 1 / sin(θ)
We can rewrite the formula as:
csc(2θ) = 1 / (1 / (1 / sin(θ)) * 1 / sec(θ))
= 1 / (1 / (csc(θ) / sec(θ)))
= sec(θ) / csc(θ)
= sec(θ) * sin(θ)
Therefore, the formula for csc(2θ) in terms of sec(θ) and csc(θ) is sec(θ) * sin(θ).