How do you find the exact value of Sin(arcsin(4/5)+arctan(12/5))sin(arcsin(45)+arctan(125))?

1 Answer
P dilip_k
Nov 8, 2016

Sin(arcsin(4/5)+arctan(12/5))sin(arcsin(45)+arctan(125))

=Sin(arcsin(4/5)+arc cot(5/12))=sin(arcsin(45)+arccot(512))

=Sin(arcsin(4/5)+arc csc(sqrt(1+(5/12)^2)=sin⎛⎝arcsin(45)+arccsc⎛⎝√1+(512)2

=Sin(arcsin(4/5)+arc csc(13/12))=sin(arcsin(45)+arccsc(1312))

=Sin(arcsin(4/5)+arc sin(12/13))=sin(arcsin(45)+arcsin(1213))

=Sin(arcsin((4/5)xxsqrt(1-(12/13)^2)+(12/13)xxsqrt(1-(4/5)^2))=sin⎛⎝arcsin⎛⎝(45)×√1−(1213)2+(1213)×√1−(45)2⎞⎠

=(4/5xx5/13)+(12/13xx3/5)=56/65=(45×513)+(1213×35)=5665

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