Question

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Proof the given identity.

cosec^2 x = 1 + cot^2 x

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Answer 1

★ Concept :-

Here the concept of Trignometric Identities has been used. We see that we are given where we have to prove that L.H.S. is equal to R.H.S. So firstly we can simplify the L.H.S. to a simpler form. We see that there are two terms in the R.H.S. So then we can simplify the R.H.S. and then bring it in the form of L.H.S. Finally then we can get our answer.

Let's do it !!

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★ Solution :-

Given to prove,

$\;\;\bf{\mapsto\;\;\red{cosec^{2}x\;=\;1\;+\;\cot^{2}x}}$

From this we get,

$\;\;\sf{\leadsto\;\;\orange{L.H.S.\;=\;cosec^{2}x}}$

$\;\;\sf{\leadsto\;\;\blue{R.H.S.\;=\;1\;+\;\cot^{2}x}}$

Now we can firstly understand different trignometric identities.

>> cosec x = 1/(sin x)

>> cot x = (cos x)/(sin x)

So let's firstly apply the value in L.H.S.

$\;\;\bf{\rightarrow\;\;L.H.S.\;=\;cosec^{2}x\;=\;\dfrac{1}{\sin^{2}x}}$

Now let's apply the value in R.H.S.

Then,

$\;\;\tt{\rightarrow\;\;R.H.S.\;=\;1\;+\;\cot^{2}x}$

$\;\;\tt{\rightarrow\;\;R.H.S.\;=\;1\;+\;\bigg(\dfrac{\cos x}{\sin x}\bigg)^{2}}$

$\;\;\tt{\rightarrow\;\;R.H.S.\;=\;1\;+\;\dfrac{\cos^{2}x}{\sin^{2}x}}$

On taking the L.C.M., we get

$\;\;\tt{\rightarrow\;\;R.H.S.\;=\;\dfrac{\sin^{2}\:+\:\cos^{2}x}{\sin^{2}x}}$

We know that,

• sin² A + cos² A = 1

This is an identity. Here A = x

By applying this here, we get

$\;\;\bf{\rightarrow\;\;\green{R.H.S.\;=\;\dfrac{1}{\sin^{2}x}}}$

Clearly, L.H.S. = R.H.S.

So,

$\;\;\bf{\rightarrow\;\;\purple{L.H.S.\;=\;R.H.S.\;=\;\dfrac{1}{\sin^{2}x}}}$

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★ More to know :-

• sec² A = 1 + tan² A

• sec A = 1/(cos A)

• cosec A = 1/(sin A)

• cot A = 1/(tan A)

• cos² A = 1 - sin² A

Answer 2

Step-by-step explanation:

$\bf\huge{\underline{\pink{ANSWER}}}$

★ Concept :-

Here the concept of Trignometric Identities has been used. We see that we are given where we have to prove that L.H.S. is equal to R.H.S. So firstly we can simplify the L.H.S. to a simpler form. We see that there are two terms in the R.H.S. So then we can simplify the R.H.S. and then bring it in the form of L.H.S. Finally then we can get our answer.

Let's do it !!

______________________________________

★ Solution :-

• Given to prove,

⭑cosec²x = 1 + cot²x

from this we get,

L.H.S. = cosec²x

R.H.S. = 1 + cot²x

Now we can firstly understand different trignometric identities.

>> cosec x = 1/(sin x)

>> cot x = (cos x)/(sin x)

So let's firstly apply the value in L.H.S.

We know that,

sin² A + cos² A = 1

This is an identity. Here A = x

________________________________

★ More to know :-

• sec² A = 1 + tan² A

• sec A = 1/(cos A)

• cosec A = 1/(sin A)

• cot A = 1/(tan A)

• cos² A = 1 - sin² A