If cot x = 5 12 for some x (, 3 2 ) , then sin 7 x ( cos 13 x 2 + sin 13 x 2 ) + cos 7 x ( cos 13 x 2 - sin 13 x 2 ) is equal to [2026]

Question

If $\cot x = \frac{5}{12}$ for some $x \in (π, \frac{3 π}{2}),$ then $\sin 7 x (\cos \frac{13 x}{2} + \sin \frac{13 x}{2}) + \cos 7 x (\cos \frac{13 x}{2} - \sin \frac{13 x}{2})$ is equal to [2026]

Smart Achievers · Accepted Answer

(1)

$\cot x = \frac{5}{12} \Rightarrow \cos x = - \frac{5}{13} = 2 \cos^{2} \frac{x}{2} - 1$

$\cos (\frac{x}{2}) = - \frac{2}{\sqrt{13}} or \frac{2}{\sqrt{13}} (rejected)$

${∵ \frac{x}{2} \in (\frac{π}{2}, \frac{3 π}{4})}$

$(\sin 7 x \cdot \frac{\sin 13 x}{2} + \cos 7 x \cdot \frac{\cos 13 x}{2}) + (\sin 7 x \cdot \frac{\cos 13 x}{2} - \cos 7 x \cdot \frac{\sin 13 x}{2})$

$= \cos (7 x - \frac{13 x}{2}) + \sin (7 x - \frac{13 x}{2})$

$= \cos \frac{x}{2} + \sin \frac{x}{2}$

$= - \frac{2}{\sqrt{13}} + \frac{3}{\sqrt{13}} = \frac{1}{\sqrt{13}}$

Solution

Q.
If $\cot x = \frac{5}{12}$ for some $x \in (π, \frac{3 π}{2}),$ then $\sin 7 x (\cos \frac{13 x}{2} + \sin \frac{13 x}{2}) + \cos 7 x (\cos \frac{13 x}{2} - \sin \frac{13 x}{2})$ is equal to [2026]

1 $\frac{1}{\sqrt{13}}$

2 $\frac{6}{\sqrt{26}}$

3 $\frac{5}{\sqrt{13}}$

4 $\frac{4}{\sqrt{26}}$

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Solution

Q. If cotx=512 for some x∈(π,3π2), then sin7x(cos13x2+sin13x2)+cos7x(cos13x2-sin13x2) is equal to [2026]

1 113

2 626

3 513

4 426