Find the minimum value of 64 tan²θ + 81 cosec²θ + 76.

Formula used :

1 + tan²θ = sec²θ 

Minimum value of asec2θ + bcosec2θ is (√a + √b)2

Calculations :

64 tan2θ + 81 cosec2θ + 76 = 64 tan2θ + 81 cosec2θ + 64 + 12

⇒ 64 + 64 tan2θ + 81 cosec2θ + 12

⇒ 64(1 + tan²θ) + 81 cosec²θ + 12 

⇒ 64 sec2θ + 81 cosec2θ + 12

Minimum value of 64 sec2θ + 81 cosec2θ = (√64 + √81)2

⇒ (8 + 9)2

⇒ 289

Now,

Minimum value of 64 sec2θ + 81 cosec2θ + 12 = 289 + 12

⇒ 301

∴ The minimum value of 64 tan2θ + 81 cosec2θ + 76 is 301.