1. Tardigrade
  2. Question
  3. Mathematics
  4. For positive integer n, if f (n) = sinn θ + cosn θ Then (f (3) - f(5)/f(5) - f(7)) is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. For positive integer $n$, if $f (n) = \sin^n \theta + cos^n \theta$
Then $\frac{f (3) - f(5)}{f(5) - f(7)}$ is

UPSEEUPSEE 2018

Solution:

Given, $f(n)=\sin ^{n} \theta+\cos ^{n} \theta \forall n \in I^{+}$
Now, $\frac{f(3)-f(5)}{f(5)-f(7)}$
$=\frac{\sin ^{3} \theta+\cos ^{3} \theta-\sin ^{5} \theta-\cos ^{5} \theta}{\sin ^{5} \theta+\cos ^{5} \theta-\sin ^{7} \theta-\cos ^{7} \theta}$
$=\frac{\sin ^{3} \theta-\sin ^{5} \theta+\cos ^{3} \theta-\cos ^{5} \theta}{\sin ^{5} \theta-\sin ^{7} \theta+\cos ^{5} \theta-\cos ^{7} \theta}$
$=\frac{\sin ^{3} \theta\left(1-\sin ^{2} \theta\right)+\cos ^{3} \theta\left(1-\cos ^{2} \theta\right)}{\sin ^{5} \theta\left(1-\sin ^{2} \theta\right)+\cos ^{5} \theta\left(1-\cos ^{2} \theta\right)}$
$=\frac{\sin ^{3} \theta \cos ^{2} \theta+\cos ^{3} \theta \sin ^{2} \theta}{\sin ^{5} \theta \cos ^{2} \theta+\cos ^{5} \theta \sin ^{2} \theta}$
$=\frac{\sin ^{2} \theta \cos ^{2} \theta(\sin \theta+\cos \theta)}{\sin ^{2} \theta \cos ^{2} \theta\left(\sin ^{3} \theta+\cos ^{3} \theta\right)}$
$=\frac{\sin ^{(1)} \theta+\cos ^{(1)} \theta}{\sin ^{(3)} \theta+\cos ^{(3)} \theta}=\frac{ f (1)}{ f (3)}$