Question

17. If cosec theta = √5 find the value of :
(i)
$2 - { \sin }^{2} theta - { \cos }^{2} theta$
(ii)
$2 + \frac{1}{ { \sin }^{2}theta } - \frac{ { \cos }^{2} theta}{ { \sin }^{2} theta}$
​

Answer 1

1 - 1ans

2- 3 ans

umeed hein apko isse samjh aa. gaya ho☺️☺️

Answer 2

Step-by-step explanation:

Given,

cosec ∅ = √5

let,

cosec∅ = √5/1

Let,

AC = √5,BC = 1,AB = ?

According to PHYTHAGAROUS THEROM,

(Hypotenuse)² = (opp.side)²+(adj.side)²

(AC)² = (AB)² + (BC)²

(√5)² = (AB)² + (1)²

5 = (AB)² + 1

5-1 = (AB)²

4 = (AB)²

√4 = AB

√2×2 = AB

:. AB = 2 units

Now consider,

Sin∅ = opp.side/Hyp = BC/AB = 1/√5

Cos∅ = adj.side/Hyp = AB/AC = 2/√5

Now consider,

(i) 2-sin²∅-cos²∅

=> 2-(1/√5)² - (2/√5)²

=> 2 - (1/5) - (4/5)

=> (10-1-4)/5

=> 5/5

=>1

:. 2-sin²∅-cos²∅ = 1

Now consider,

(ii) 2+(1/sin²∅)-(cos²∅/sin²∅)

=> 2 + (1-cos²∅)/sin²∅

=> 2 + [1-(2/√5)²]/(1/√5)²

=> 2 + [1-(4/5)]/(1/5)

=> 2 + [(5-4)/5]/(1/5)

=> 2 + (1/5)/(1/5)

=> 2 + (1/5 × 5/1)

=> 2 + (5/5)

=> 2 + 1

=> 3

:. 2+(1/sin²∅)-(cos²∅/sin²∅) = 3.

I hope it helps you.