Question

Sin theta+cosec theta=2 then sin^n theta+cosec^n theta

Answer 1

Answer:

$sin^n \ \theta + cosec^n \ \theta = 2$

Step-by-step explanation:

Given :

$sin \ \theta + cosec \ \theta = 2$

To find :

The value of :

$sin^n \ \theta + cosec^n \ \theta$

Solution :

$sin \ \theta + cosec \ \theta = 2$

⇒ $sin \ \theta + \frac{1}{sin \ \theta} = 2$

⇒ $\frac{sin^2 \ \theta+1}{sin \ \theta}=2$

⇒ $sin^2 \ \theta+1=2 \ sin \ \theta$

⇒ $sin^2 \ \theta+1-2 \ sin \ \theta=0$

⇒ $sin \ \theta(sin \ \theta - 1)-1(sin \ \theta - 1) = 0$

⇒ $(sin \ \theta - 1)^2=0$

⇒ $sin \ \theta -1=0$

⇒ $sin \ \theta = 1$

⇒ $cosec \ \theta = \frac{1}{sin \ \theta} = \frac{1}{1}=1$

⇒ $sin^n \ \theta = 1$             (As, $1^n = 1$)

⇒ $cosec^n \ \theta = 1$

Hence,

⇒ $sin^n \ \theta + cosec^n \ \theta = 1 + 1 = 2$