Prove that: sin−1(53)+cos−1(1312)=sin−1(6556).

Let cos−1(1312)=x⇒1312=cosx∴sinx=135Let sin−1(53)=y⇒53=siny∴cosy=54We know that, sin(x+y)=sinxcosy+cosxsinysin(x+y)=135×54+1312×53sin(x+y)=6556∴x+y=sin−1(6556)cos−1(1312)+sin−1(53)=sin−1(6556)Hence proved.

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Prove that: ^{-1} ( {3}{5} ) + ^{-1} ( {12}{13} ) = ^{-1} ( {

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Prove that: $sin^{- 1} (\frac{3}{5}) + cos^{- 1} (\frac{12}{13}) = sin^{- 1} (\frac{56}{65})$ .

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Prove that:

sin^{- 1} (\frac{3}{5}) + cos^{- 1} (\frac{12}{13}) = sin^{- 1} (\frac{56}{65})

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Text solutionVerified

Let

cos^{- 1} (\frac{12}{13}) = x

\Rightarrow \frac{12}{13} = cos x

∴ sin x = \frac{5}{13}

Let

sin^{- 1} (\frac{3}{5}) = y

\Rightarrow \frac{3}{5} = sin y

∴ cos y = \frac{4}{5}

We know that,

sin (x + y) = sin x cos y + cos x sin y

sin (x + y) = \frac{5}{13} \times \frac{4}{5} + \frac{12}{13} \times \frac{3}{5}

sin (x + y) = \frac{56}{65}

∴ x + y = sin^{- 1} (\frac{56}{65})

cos^{- 1} (\frac{12}{13}) + sin^{- 1} (\frac{3}{5}) = sin^{- 1} (\frac{56}{65})

Hence proved.

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Question Text	Prove that: $sin^{- 1} (\frac{3}{5}) + cos^{- 1} (\frac{12}{13}) = sin^{- 1} (\frac{56}{65})$ .
Answer Type	Text solution:1
Upvotes	150

Prove that: sin−1(53​)+cos−1(1312​)=sin−1(6556​).

Text solutionVerified

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Prove that: $sin^{- 1} (\frac{3}{5}) + cos^{- 1} (\frac{12}{13}) = sin^{- 1} (\frac{56}{65})$ .