Question

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

• A. $\frac{f(1)}{f(3)}$
• B. $\frac{f(3)}{f(1)}$
• C. $\frac{f(3)}{f(5)}$
• D. $\frac{f(5)}{f(7)}$

Answer 1

Answer: (A)

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