Question

If cosec theta - sin theta = m and sec theta - cos theta = n , prove that (m 2 n) 2/3 + (mn ​2 ) 2/3 = 1

Answer 1

Solution:-
Given cosec theta - sin theta = m, sec theta - cos theta = n
Given that cosec theta - sin theta = m
→ !/sin theta - sin theta = m
⇒ (1-sin² theta)/sin theta = m → cos² theta/sin theta = m
and sec theta - cos theta = n
⇒ 1/cos theta - cos theta = n → (1-cos² theta)/cos theta = n
sin² theta/cos theta = n
Now (m²n)²/³ + (mn²)²/³
⇒ (cos⁴ theta/sin² theta × sin² theta/cos theta)²/³ + (cos² theta/sin theta × sin⁴ theta/cos² theta)²/³
⇒ (cos³ theta)²/³ + (sin³ theta)²/³
⇒cos² theta + sin² theta
= 1 Hence proved

Answer 2

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