Question

no bul.lshit plz , answer only if u know how to solve

Answer 1

Given Tn = sin^ntheta + cos^ntheta.

LHS:

T3 - T5/T1

= (sin^3 theta + cos^3theta) - (sin^5theta + cos^5theta)/sintheta+costheta

= (sin^3theta + cos^3theta - sin^5theta - cos^5theta)/sintheta+costheta

= sin^3theta(1 - sin^2theta) + cos^3theta(1 - cos^2theta)/sintheta + costheta

We know that 1 - sin^2 theta = cos^2 theta & 1 - cos^2 theta = sin^2 theta

= sin^3theta * cos^2theta + cos^3theta * sin^2theta/sin theta + cos theta

= sin^2 theta cos^2 theta(sin theta + cos theta)/sin theta + cos theta)

= sin^2 theta cos^2 theta.



RHS :

T5 - T7/T3

= (sin^5 theta + cos^5 theta) - (sin^7theta + cos^7theta)/(sin ^3 theta + cos^3 theta)

= (sin^5 theta + cos^ 5theta - sin^7 theta - cos^ 7theta)/(sin^3 theta + cos^3 theta)

= sin^5theta(1 - sin^2theta)  + cos^5 theta(1 - cos^2 theta)/(sin^3 theta + cos^3 theta)

= sin^5 theta * cos^2 theta + cos^5 theta * sin^2 theta/(sin^3 theta + cos^3 theta)

= sin^2 theta cos^2 theta(sin^3 theta + cos^3 theta)/(sin^3 theta + cos^3 theta)

= sin^2 theta * cos^2 theta   


Therefore LHS = RHS.


Hence Proved.


Hop e this helps!