Question

Prove the following trigonometric identities:
cos²A+1/1+cot²A=1

Answer 1

Step-by-step explanation:

To prove : ${\sf{\ \ cos^2 A + {\dfrac{1}{1 + cot^2 A}} = 1}}$

Proof :

L.H.S. = ${\sf{\ \ cos^2 A + {\dfrac{1}{1 + cot^2 A}}}}$

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${\boxed{\sf{\red{Identity \ : \ 1 + cot^2 \theta = cosec^2 \theta}}}}$

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$\Rightarrow{\sf{cos^2 A + {\dfrac{1}{cosec^2 A}}}}$

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We can write this as :

$\Rightarrow{\sf{cos^2 A + {\dfrac{(1)^2}{(cosec A)^2}}}}$

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$\Rightarrow{\sf{cos^2 A + \left( {\dfrac{1}{cosec A}} \right) ^2}}$

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

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$\Rightarrow{\sf{ cos^2 A + (sin A)^2 }}$

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$\Rightarrow{\sf{ cos^2 A + sin^2 A}}$

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

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

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= R.H.S.

Hence, proved !!

Answer 2

$\bf To\ prove:\ \tt cos^2\ A\ +\ \dfrac{1}{1\ +\ cot^2\ A}\ =\ 1.\\\\\\\bf Proof:\\\\\\\bf Left\ hand\ side:\\\\\\\tt cos^2\ A\ +\ \dfrac{1}{1\ +\ cot^2\ A}\\\\\\\sf We\ know\ that\ cosec^2\ \theta\ -\ cot^2\ \theta\ =\ 1.\\\\Therefore, 1\ +\ cot^2\ \theta\ = \ cosec^2\ \theta.\\\\\\\tt cos^2\ A\ +\ \dfrac{1}{cosec^2\ A}\\\\\\\sf We\ know\ that\ sin\ \theta\ =\ \dfrac{1}{cosec\ \theta}.\\\\Therefore, \dfrac{1}{cosec^2\ \theta}\ =\ sin^2\ \theta.\\\\\\\tt cos^2\ A\ +\ sin^2\ A$

= 1

= Right hand side.

Hence proved.