Write the following as a single sine or cosine function of an angle: \( (\sin 7 \theta+\cos 7 \theta)^{2}-1 \)

To rewrite \( (\sin 7\theta + \cos 7\theta)^2 - 1 \) as a single sine or cosine function, we start by expanding the expression: \[ (\sin 7\theta + \cos 7\theta)^2 = \sin^2 7\theta + 2\sin 7\theta \cos 7\theta + \cos^2 7\theta \] Using the Pythagorean identity \( \sin^2 x + \cos^2 x = 1 \): \[ \sin^2 7\theta + \cos^2 7\theta = 1 \] So, we can simplify the expansion: \[ (\sin 7\theta + \cos 7\theta)^2 = 1 + 2\sin 7\theta \cos 7\theta \] Now, using the double angle identity \( 2\sin x \cos x = \sin(2x) \), we can further simplify: \[ (\sin 7\theta + \cos 7\theta)^2 = 1 + \sin(14\theta) \] Now we incorporate the subtraction from the original expression: \[ (\sin 7\theta + \cos 7\theta)^2 - 1 = \sin(14\theta) \] Thus, the final result is: \[ \sin(14\theta) \] So, \( (\sin 7 \theta+\cos 7 \theta)^2-1 \) can be expressed as \( \sin(14\theta) \).