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Conv. trigonometric function identities: $\cot^2{(\pi/8)}+1=3\sec{(\pi/8)}.$
Can someone explain how did they get this:
$\cot^2{(\pi/8)}+1=3\sec{(\pi/8)}.$
$$\frac{1}{4\sin^2{(\pi/8)}}+1=\frac{3}{4}\sec^2{(\pi/8)}$$
I am confused as to how they got $3\sec{(\pi/8)}$. In this case $\cot^2{(\pi/8)}+1=3\sec{(\pi/8)}$
A:
The identity in question is seen to be true by plotting the numerator and denominator of the left hand side of the identity.
The plot of the numerator of the left hand side is the red line. The plot of the denominator of the left hand side is the blue line.
The numerator is seen to be as shown, when the blue line is rotated by $\frac{ -2\pi}{8}$ radians, or $\frac{ -\pi}{8}$ radians, and as seen, when the red line is also rotated by $\frac{ -\pi}{8}$ radians, or $\frac{\pi}{8}$ radians.
The difference in the angles of the rotated blue and red lines is $\frac{\pi}{8}$ radians, or $\frac{3\pi}{8}$ radians, and it follows that:
$$\cot^2 \left(\frac{\pi}{8}\right)+1=3\sec
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