Given the equation n=(sin(theta+alpha))/(sin theta), where alpha=45^(@), then an equivalent expression for n is

Expand using angle addition formula: Use the angle addition formula for sine to expand sin(theta + alpha). The angle addition formula for sine is sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Here, A = theta and B = alpha = 45 degrees. sin(theta + 45 degrees) = sin(theta)cos(45 degrees) + cos(theta)sin(45 degrees).Substitute values and simplify: Substitute the values of sin(45 degrees) and cos(45 degrees). Since sin(45 degrees) = cos(45 degrees) = sqrt(2)/2, we can substitute these values into the expanded expression. sin(theta + 45 degrees) = sin(theta)(sqrt(2)/2) + cos(theta)(sqrt(2)/2).Substitute into equation for n: Substitute the expanded expression of sin(theta + 45 degrees) into the equation for n. n = (sin(theta + 45 degrees)) / (sin theta) n = (sin(theta)(sqrt(2)/2) + cos(theta)(sqrt(2)/2)) / (sin theta).Simplify numerator terms: Simplify the expression by dividing each term in the numerator by sin(theta). n = (sin(theta)(sqrt(2)/2) / sin(theta)) + (cos(theta)(sqrt(2)/2) / sin(theta)) n = (sqrt(2)/2) + (cos(theta)/sin(theta))(sqrt(2)/2).Use cotangent identity: Recognize that cos(theta)/sin(theta) is the cotangent of theta. cot(theta) = cos(theta)/sin(theta). Substitute cot(theta) into the expression. n = (sqrt(2)/2) + cot(theta)(sqrt(2)/2).Factor out common factor: Factor out the common factor of sqrt(2)/2. n = (sqrt(2)/2)(1 + cot(theta)).

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Given the equation
n=(sin(theta+alpha))/(sin theta), where
alpha=45^(@), then an equivalent expression for
n is

Given the equation $n=\frac{\sin (\theta+\alpha)}{\sin \theta}$ , where $\alpha=45^{\circ}$ , then an equivalent expression for $n$ is

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Q. Given the equation

n=\frac{\sin (\theta+\alpha)}{\sin \theta}

, where

\alpha=45^{\circ}

, then an equivalent expression for

n

Expand using angle addition formula: Use the angle addition formula for sine to expand $\sin(\theta + \alpha)$ . The angle addition formula for sine is $\sin(A + B) = \sin(A)\cos(B) + \cos(A)\sin(B)$ . Here, $A = \theta$ and $B = \alpha = 45$ degrees. $\sin(\theta + 45$ degrees) = $\sin(\theta)\cos(45$ degrees) + $\cos(\theta)\sin(45$ degrees).
Substitute values and simplify: Substitute the values of $\sin(45^\circ)$ and $\cos(45^\circ)$ . Since $\sin(45^\circ) = \cos(45^\circ) = \sqrt{2}/2$ , we can substitute these values into the expanded expression. $\sin(\theta + 45^\circ) = \sin(\theta)(\sqrt{2}/2) + \cos(\theta)(\sqrt{2}/2)$ .
Substitute into equation for $n$ : Substitute the expanded expression of $\sin(\theta + 45^\circ)$ into the equation for $n$ .
$n = \frac{\sin(\theta + 45^\circ)}{\sin \theta}$
$n = \frac{\sin(\theta)\left(\frac{\sqrt{2}}{2}\right) + \cos(\theta)\left(\frac{\sqrt{2}}{2}\right)}{\sin \theta}$ .
Simplify numerator terms: Simplify the expression by dividing each term in the numerator by $\sin(\theta)$ . $\newline$ $n = \left(\frac{\sin(\theta)\left(\frac{\sqrt{2}}{2}\right)}{\sin(\theta)}\right) + \left(\frac{\cos(\theta)\left(\frac{\sqrt{2}}{2}\right)}{\sin(\theta)}\right)$ $\newline$ $n = \left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\cos(\theta)}{\sin(\theta)}\right)\left(\frac{\sqrt{2}}{2}\right)$ .
Use cotangent identity: Recognize that $\cos(\theta)/\sin(\theta)$ is the cotangent of $\theta$ . $\newline$ $\cot(\theta) = \cos(\theta)/\sin(\theta)$ . $\newline$ Substitute $\cot(\theta)$ into the expression. $\newline$ $n = (\sqrt{2}/2) + \cot(\theta)(\sqrt{2}/2)$ .
Factor out common factor: Factor out the common factor of $\frac{\sqrt{2}}{2}$ . $\newline$ $n = \left(\frac{\sqrt{2}}{2}\right)(1 + \cot(\theta))$ .