Question

Prove the following trigonometric identities:
cosecθ√1-cos²θ=1

Answer 1

Step-by-step explanation:

To Prove : ${\sf{\ \ cosec \theta {\sqrt{1 - cos^2 \theta}} = 1}}$

L.H.S. = ${\sf{\ cosec \theta {\sqrt{1 - cos^2 \theta}}}}$

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${\boxed{\sf{\red{Identity \ : \ sin^2 \theta + cos^2 \theta = 1}}}}$

${\sf{\red{From \ this, \ we \ get \ sin^2 \theta = 1 - cos^2 \theta}}}$

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$\Rightarrow{\sf{cosec \theta {\sqrt{sin^2 \theta}}}}$

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We can write this as :

$\Rightarrow{\sf{cosec \theta {\sqrt{(sin \theta)^2}}}}$

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If we remove a square root from a number, then the number is raised to the power of 1/2.

$\Rightarrow{\sf{cosec \theta \times [ (sin \theta)^2 ] ^{\frac{1}{2}} }}$

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${\boxed{\sf{\red{Identity \ : \ (a^m)^n = (a)^{mn} }}}}$

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$\Rightarrow{\sf{cosec \theta \times (sin \theta)^{2 \times {\frac{2}{2}} } }}$

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$\Rightarrow{\sf{cosec \theta \times sin \theta}}$

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${\boxed{\sf{\red{Identity \ : \ cosec \theta = {\dfrac{1}{sin \theta}} }}}}$

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$\Rightarrow{\sf{ {\dfrac{1}{sin \theta}} \times sin \theta}}$

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$\Rightarrow{\boxed{\sf{\green{1}}}}$

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= R.H.S.

Hence, proved !!

Answer 2

Answer :

To prove :

$cosec \alpha \sqrt{1 - {cos}^{2} \alpha } = 1$

Proof :

L. H. S

$cosec \alpha \sqrt{1 - {cos}^{2} \alpha }$

As the Trigonometric identity,

Sin ² a + cos ² a =1

Sin ² a = 1- cos ² a

Therfore by using this identity,

$= > cosec \alpha \sqrt{ {sin}^{2} \alpha }$

$= > cosec \alpha \times sin \alpha$

$as \: cosec \: \alpha = \frac{1}{sin \: \alpha }$

So,

$= > \frac{1}{sin \alpha } \times sin \alpha$

$= > \: 1$

As R. H. S =1.

Therefore L. H. S=R. H. S.

Hence proved.