Evaluate tan-1-2sin-12a-1-a2-1-2cos-11-a2-1-a2-

Explanation for the correct option:Evaluating the given trigonometric expressionGiven, tan[(1/2)sin^ 1(2a/1+a^2)+(1/2)cos^ 1(1 a^2/1+a^2)]We know the Trigonome ...

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Standard XII

Mathematics

Particular Solution of a Differential Equation

Evaluate tan[...

Question

Evaluate $\tan [(\frac{1}{2}) \sin^{- 1} (\frac{2 a}{1 + a^{2}}) + (\frac{1}{2}) \cos^{- 1} (\frac{1 - a^{2}}{1 + a^{2}})]$ =

$\frac{2 a}{1 + a^{2}}$

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$\frac{1 - a^{2}}{1 + a^{2}}$

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$\frac{2 a}{1 - a^{2}}$

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None of these

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Solution

The correct option is C
$\frac{2 a}{1 - a^{2}}$

Explanation for the correct option:
Evaluating the given trigonometric expression
Given, $\tan [(\frac{1}{2}) \sin^{- 1} (\frac{2 a}{1 + a^{2}}) + (\frac{1}{2}) \cos^{- 1} (\frac{1 - a^{2}}{1 + a^{2}})]$
We know the Trigonometric Identity,
$\sin^{- 1} (\frac{2 x}{1 + x^{2}}) = 2 \tan^{- 1} x \to (i)$
$\cos^{- 1} (\frac{1 - x^{2}}{1 + x^{2}}) = 2 \tan^{- 1} x \to (i i)$
$\tan (2 θ) = \frac{2 \tan θ}{1 - \tan^{2} θ} \to (i i i)$
Using the above identities in the given expression we get,
$\tan [(\frac{1}{2}) \sin^{- 1} (\frac{2 a}{1 + a^{2}}) + (\frac{1}{2}) \cos^{- 1} (\frac{1 - a^{2}}{1 + a^{2}})] = \tan [(\frac{1}{2}) 2 \tan^{- 1} a + (\frac{1}{2}) 2 \tan^{- 1} a] [from (i) and (i i)] = \tan [2 \tan^{- 1} (a)] = \frac{2 \tan (\tan^{- 1} a)}{1 - {(\tan (\tan^{- 1} a))}^{2}} [from (i i i)] = \frac{2 a}{1 - a^{2}}$
Therefore, we get,
$\Rightarrow \tan [(\frac{1}{2}) \sin^{- 1} (\frac{2 a}{1 + a^{2}}) + (\frac{1}{2}) \cos^{- 1} (\frac{1 - a^{2}}{1 + a^{2}})] = \frac{2 a}{1 - a^{2}}$
Hence, the correct answer is Option (C).

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Similar questions

From the following place value table, write the decimal number:-

Thousand	Hundred	Tens	Ones	Tenths	Hundredths	Thousandths
$(1000)$	$(100)$	$(10)$	$(1)$	$(\frac{1}{10})$	$(\frac{1}{100})$	$(\frac{1}{1000})$
			$9$	$6$

From the given place value table, write the decimal number.

Thousands	Hundreds	Tens	Ones	Tenths	Hundredths	Thousandths
$(1000)$	$(100)$	$(10)$	$(1)$	$(\frac{1}{10})$	$(\frac{1}{100})$	$(\frac{1}{1000})$
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Q. Evaluate the following:
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{(\sqrt{x + 1} + \sqrt{x - 1})}^{6} + {(\sqrt{x + 1} - \sqrt{x - 1})}^{6}

(ii)

{(x + \sqrt{x^{2} - 1})}^{6} + {(x - \sqrt{x^{2} - 1})}^{6}

(iii)

{(1 + 2 \sqrt{x})}^{5} + {(1 - 2 \sqrt{x})}^{5}

(iv)

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{(3 + \sqrt{2})}^{5} - {(3 - \sqrt{2})}^{5}

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(x)

{\{a^{2} + \sqrt{a^{2} - 1}\}}^{4} + {\{a^{2} - \sqrt{a^{2} - 1}\}}^{4}

Evaluate the following:

$(i) (\sqrt{x + 1} + \sqrt{x - 1})^{6} + (\sqrt{x + 1} - \sqrt{x - 1})^{6}$

$(i i) (x + \sqrt{x^{2} - 1})^{6} + (x - \sqrt{x^{2} - 1})^{6}$

$(i i i) (1 + 2 \sqrt{x})^{5} + (1 - 2 \sqrt{x})^{5}$

$(i v) (\sqrt{2} + 1)^{6} + (\sqrt{2} - 1)^{6}$

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$(v i) (2 + \sqrt{3})^{7} + (2 - \sqrt{3})^{7}$

$(v i i) (\sqrt{3} + 1)^{5} - (\sqrt{3} - 1)^{5}$

$(v i i i) (0.99)^{5} + (1.01)^{5}$

$(i x) (\sqrt{3} + \sqrt{2})^{6} - (\sqrt{3} - \sqrt{2})^{6}$

$(x) {a^{2} + \sqrt{a^{2} - 1}}^{4} + {a^{2} - \sqrt{a^{2} - 1}}^{4}$

Q. If

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, find x and y such that

A^{2} = x A + y I = O

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Evaluate tan12sin-12a1+a2+12cos-11-a21+a2=

Evaluate $\tan [(\frac{1}{2}) \sin^{- 1} (\frac{2 a}{1 + a^{2}}) + (\frac{1}{2}) \cos^{- 1} (\frac{1 - a^{2}}{1 + a^{2}})]$ =