7. given that \\( \\tan \\frac { 7 \\pi } { 8 } = - \\cot z \\), first express \\( \\frac { 7 \\pi } { 8 }…

7. given that \\( \\tan \\frac { 7 \\pi } { 8 } = - \\cot z \\), first express \\( \\frac { 7 \\pi } { 8 } \\) as a sum of \\( \\frac { \\pi } { 2 } \\) and an angle, and then apply a trigonometric identity to determine the measure of angle \\( z \\).

7. given that \\( \\tan \\frac { 7 \\pi } { 8 } = - \\cot z \\), first express \\( \\frac { 7 \\pi } { 8 } \\) as a sum of \\( \\frac { \\pi } { 2 } \\) and an angle, and then apply a trigonometric identity to determine the measure of angle \\( z \\).

Answer

Explanation:

Step1: Express (\frac{7\pi}{8}) as a sum

We know that (\frac{7\pi}{8}=\frac{\pi}{2}+\frac{3\pi}{8}).

Step2: Apply the trigonometric identity

We use the identity (\tan(A + B)=\frac{\tan A+\tan B}{1 - \tan A\tan B}), but here we can also use the co - function identity (\tan\left(\frac{\pi}{2}+\alpha\right)=-\cot\alpha). Since (\tan\frac{7\pi}{8}=\tan\left(\frac{\pi}{2}+\frac{3\pi}{8}\right)), and by the identity (\tan\left(\frac{\pi}{2}+\alpha\right)=-\cot\alpha), when (A = \frac{\pi}{2}) and (\alpha=\frac{3\pi}{8}), we have (\tan\left(\frac{\pi}{2}+\frac{3\pi}{8}\right)=-\cot\frac{3\pi}{8})

Given (\tan\frac{7\pi}{8}=-\cot z), comparing with (\tan\frac{7\pi}{8}=-\cot\frac{3\pi}{8})

Answer:

(z = \frac{3\pi}{8})