Question

If sin θ + cosec θ = 2 , then find sin^19 θ + cosec^19 θ

Answer 1

Step-by-step explanation:

$sinθ + cosecθ = 2$

$sinθ+ \frac{1}{sinθ} =2$

$\frac{sin²θ + 1}{sinθ} =2$

$sin²θ+1=2sinθ$

$sin²θ-2sinθ+1=0$

$sin²θ-sinθ-sinθ+1$

$sinθ(sinθ-1)-1(sinθ-1)=0$

$(sinθ-1)(sinθ-1)$

$(sinθ-1)²=0$

$sinθ-1=0$

$sinθ=1$

$θ=90°$

$now$

$sin¹⁹θ+cosec¹⁹θ$

$=sin¹⁹(90)+cosec¹⁹(90)$

$=(1)¹⁹+(1)¹⁹$

$=1+1$

$sin¹⁹θ+cosec¹⁹θ=2$