Solve
#sin^-1((ax)/c) + sin^-1( (bx)/c) = sin^-1( x )#where a²+b²=c² and c≠0
#sin^-1((ax)/c) + sin^-1( (bx)/c) = sin^-1( x )#
#=>sin^-1{((ax)/c)sqrt(1-( (bx)/c)^2)+( (bx)/c)sqrt(1-( (ax)/c)^2)} = sin^-1( x )#
#=>{((ax)/c)sqrt(1-( (bx)/c)^2)+( (bx)/c)sqrt(1-( (ax)/c)^2)}^2 = x^2 #
#=>((ax)/c)^2(1-( (bx)/c)^2)+( (bx)/c)^2(1-( (ax)/c)^2)+2((ax)/c)sqrt(1-( (bx)/c)^2) ((bx)/c)sqrt(1-( (ax)/c)^2) = x^2 #
#=>((a^2+b^2))/c^2x^2-(2a^2b^2x^4)/c^4+2((ax)/c)sqrt(1-( (bx)/c)^2) ((bx)/c)sqrt(1-( (ax)/c)^2) = x^2 #
#=>c^2/c^2x^2-(2a^2b^2x^4)/c^4+2((ax)/c)sqrt(1-( (bx)/c)^2) ((bx)/c)sqrt(1-( (ax)/c)^2) = x^2 #
#=>2((ax)/c)sqrt(1-( (bx)/c)^2) ((bx)/c)sqrt(1-( (ax)/c)^2) = (2a^2b^2x^4)/c^4 #
#=>x^2sqrt(1-( (bx)/c)^2) sqrt(1-( (ax)/c)^2) - (abx^4)/c^2=0 #
#=>x^2[sqrt(1-( (bx)/c)^2) sqrt(1-( (ax)/c)^2) - (abx^2)/c^2]=0 #
so #x^2=0=>x=0#
And
#=>sqrt(1-( (bx)/c)^2) sqrt(1-( (ax)/c)^2)= (abx^2)/c^2 #
#=>1-((a^2+b^2))/c^2x^2+cancel((a^2b^2x^4)/c^2)= cancel((a^2b^2x^4)/c^2#
#=>x^2-1=0#
#=>x=+-1#
Ans : #x=0,+1 and -1#
