Paragraph: Read the following information and answer Let t_n= ^n θ+ ^n θ . Question: What is (t_3-t_5)/(t_5-t_7) equal to?

Correct answer: A. (t_1)/(t_3) t_n= ^n θ+ ^n θ Now, (t_3-t_5)/(t_5-t_7)=( ^3 θ+ ^3 θ- ^5 θ- ^5 θ)/( ^5 θ+ ^5 θ- ^7 θ- ^7 θ) aligned & =( ^3 θ(1- ^2 θ)+ ^3 θ(1- ^2 θ))/( ^5 θ(1- ^2 θ)+ ^7 θ(1- ^2 θ)) & =( ^3 θ · ^2 θ+ ^3 θ · ^2 θ)/( ^5 θ · ^2 θ+ ^7 θ · ^2 θ) aligned aligned & =( ^2 θ · ^2 θ( θ+ θ))/( ^2 θ · ^2 θ( ^3 θ+ ^3 θ)) & =(( θ+ θ))/(( ^3 θ+ ^3 θ))=(t_1)/(t_3) aligned

NDA 2020

Exam: NDA
Year: 2020
Subject: Mathematics
Chapter: Trigonometric Ratios & Identities

NDA 2020 · Mathematics · Trigonometric Ratios & Identities

Paragraph: Read the following information and answer Let t_n= ^n θ+...

Paragraph: Read the following information and answer Let

t_n=\sin ^n \theta+\cos ^n \theta

.
Question: What is

\frac{t_3-t_5}{t_5-t_7}

equal to?

Options

A $\frac{t_1}{t_3}$
B $\frac{t_3}{t_5}$
C $\frac{t_5}{t_7}$
D $\frac{t_1}{t_7}$

Correct answer

A. $\frac{t_1}{t_3}$

Step-by-step solution

t_n=\sin ^n \theta+\cos ^n \theta

Now,

\frac{t_3-t_5}{t_5-t_7}=\frac{\sin ^3 \theta+\cos ^3 \theta-\sin ^5 \theta-\cos ^5 \theta}{\sin ^5 \theta+\cos ^5 \theta-\sin ^7 \theta-\cos ^7 \theta}

\begin{aligned} & =\frac{\sin ^3 \theta\left(1-\sin ^2 \theta\right)+\cos ^3 \theta\left(1-\cos ^2 \theta\right)}{\sin ^5 \theta\left(1-\sin ^2 \theta\right)+\cos ^7 \theta\left(1-\cos ^2 \theta\right)} \\ & =\frac{\sin ^3 \theta \cdot \cos ^2 \theta+\cos ^3 \theta \cdot \sin ^2 \theta}{\sin ^5 \theta \cdot \cos ^2 \theta+\cos ^7 \theta \cdot \sin ^2 \theta}\end{aligned}

$\begin{aligned} & =\frac{\sin ^2 \theta \cdot \cos ^2 \theta(\sin \theta+\cos \theta)}{\sin ^2 \theta \cdot \cos ^2 \theta\left(\sin ^3 \theta+\cos ^3 \theta\right)} \\ & =\frac{(\sin \theta+\cos \theta)}{\left(\sin ^3 \theta+\cos ^3 \theta\right)}=\frac{t_1}{t_3}\end{aligned}

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