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The value of sin (tan ^-13/4+tan ^-15/12) is Single correct option
The value of
sin ( tan − 1 3 4 + tan − 1 5 12 ) \sin\left(\tan^{-1}\dfrac34+\tan^{-1}\dfrac5{12}\right) sin ( tan − 1 4 3 + tan − 1 12 5 ) is
A. 56 65 \dfrac{56}{65} 65 56 B. 33 65 \dfrac{33}{65} 65 33 C. 63 65 \dfrac{63}{65} 65 63 D. 16 65 \dfrac{16}{65} 65 16 Show answer and solution Answer: (A) 56 65 \dfrac{56}{65} 65 56
Using
sin ( A + B ) = sin A cos A … \sin(A+B)=\sin A\cos A\ldots sin ( A + B ) = sin A cos A … directly:
sin A = 3 5 , cos A = 4 5 \sin A=\tfrac35,\cos A=\tfrac45 sin A = 5 3 , cos A = 5 4 and
sin B = 5 13 , cos B = 12 13 \sin B=\tfrac5{13},\cos B=\tfrac{12}{13} sin B = 13 5 , cos B = 13 12 . So
sin ( A + B ) = 3 5 ⋅ 12 13 + 4 5 ⋅ 5 13 = 36 + 20 65 = 56 65 \sin(A+B)=\tfrac35\cdot\tfrac{12}{13}+\tfrac45\cdot\tfrac5{13}=\tfrac{36+20}{65}=\tfrac{56}{65} sin ( A + B ) = 5 3 ⋅ 13 12 + 5 4 ⋅ 13 5 = 65 36 + 20 = 65 56 . Option
33 65 \tfrac{33}{65} 65 33 is
cos ( A + B ) \cos(A+B) cos ( A + B ) , found by mixing up the signs.
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