Question

1.) Show that none of the following is an identity :
I. cos² 0 + cos 0 = 1
ii. sin²0 + sin 0 = 2
iii. tan ² 0 + sin 0 = cos²0

2.)
$\frac{cos \: 0 \: cosec \: 0 \: - \: sin \: 0 \: sec \: 0}{cos \: 0 \: + \: sin \: 0} = cosec \: 0 - sec \: 0$
PLEASE ANY ONE SOLVE THIS QUESTION WITH EXPLANATION...​

Answer 1

Step-by-step explanation:

To proof none of them is an identity, we need to put a value in the place of θ so that we can compare the LHS and RHS.

Thus, let's say θ = 60°.

Hence,

I. cos² θ + cos θ = 1

cos² 60 + cos 60 = 1;

(1/2)^2 + (1/2) = 1;

1/4 + 1/2 = 1;

3/4 ≠ 1;

Thus, cos² θ +cos θ= 1, is not an identity because it should've worked for every value (0 ≤ θ ≥ 90) of θ.

Here, let θ be 30°.

Hence,

II. sin² θ + sin θ = 2;

sin² 30 + sin 30 = 2;

(1/2)^2 + (1/2) = 2;

1/4 + 1/2 = 2;

3/4 ≠ 2;

Thus, sin² θ +sin θ= 2 is not an identity because it should've worked for every value (0 ≤ θ ≥ 90) of θ.

Here, let θ be 0°.

Hence,

III. tan ² θ + sin θ = cos² θ;

tan ² 0 + sin 0 = cos² 0;

0 + 0 = 1.

0 ≠ 1.

Thus, tan² θ +sin θ=cos² θ, is not an identity because it should've worked for every value (0 ≤ θ ≥ 90) of θ.

2. (cos θ.cosec θ - sin θ.secθ)/(cos θ +sin θ) = cosec θ - sec θ;

= (cos θ.(1/sin θ) - sin θ.(1/cosθ))/(cos θ +sin θ) = (1/sin θ) - (1/cos θ);

= (cos θ/sin θ - sin θ/ cos θ)/(cos θ +sin θ) = (cos θ - sin θ)/cos θ.sinθ;

= ((cos^2 θ - sin^2 θ)/sinθ.cos θ)/(cos θ +sin θ) = (cos θ - sin θ)/cos θ.sinθ;

= ((cos^2 θ - sin^2 θ))/(sinθ.cos θ)(cos θ +sin θ) = (cos θ - sin θ)/cos θ.sinθ;

= ((cos^2 θ - sin^2 θ))/(cos θ +sin θ) = (cos θ - sin θ);

= ((cos^2 θ - sin^2 θ))/(cos θ +sin θ) = (cos θ - sin θ);

= ((cos θ - sin θ)(cos θ +sin θ))/(cos θ +sin θ) = (cos θ - sin θ); (since, a^2 - b^2 =(a+ b)(a -b))

= (cos θ - sin θ) = (cos θ - sin θ);

Thus. LHS = RHS.

That's all.

Answer 2

2.) cos theta × cosec theta- sin theta × sec theta/ cos theta + sin theta = cosec theta- sec theta

Step- by- step explanation:-

Ans) cos theta ×1/sin theta- sin theta ×1/cos theta÷ cos theta + sin theta

cos theta/ sin theta- sin theta/ cos theta ÷ cos theta + sin theta

cos ^2 theta- sin ^2 theta/ sin theta × cos theta ÷ cos theta +sin theta/1

cos ^2 theta- sin ^2theta/sin theta. cos theta ×1/cos theta + sin theta

cos ^2 theta- sin ^2/sin theta. cos theta (cos theta +sin theta)

(cos theta- sin theta)(cos theta +sin theta )/ sin theta. cos theta (cos theta +sin theta)

(cos theta- sin theta)/sin theta. cos theta

cos theta/ sin theta. cos theta- sin theta/ sin theta. cos theta

1/sin theta- 1/cos theta

cosec theta- sec theta . R.H.S