prove cscx/1+cscx + cscx/ 1-cscx = -2sinx/cos^2x?
Hi, need to prove this, and I've tried what feels like a thousand ways and I keep getting stuck.
so if someone could help that would be great thanks.
cscx/1+cscx + cscx/ 1-cscx = -2sinx/cos^2x
Hi, need to prove this, and I've tried what feels like a thousand ways and I keep getting stuck.
so if someone could help that would be great thanks.
cscx/1+cscx + cscx/ 1-cscx = -2sinx/cos^2x
Comments
cosecx(1+cosecx) + cosecx/(1-cosecx) ≡ -2sinx/cos² x
Left hand side = cosecx/(1+cosecx) + cosecx/(1-cosecx)
=> (1/sinx) /( 1 + (1/sinx) )+ (1/sinx)/ ( 1-(1/sinx) )
=>(1/sinx) / ( (sinx+1) /sinx ) + (1/sinx) / ( (sinx-1) /sinx)
=>(1/sinx) ( sinx / (sinx+1) ) + (1/sinx) (sinx/ (sinx-1) )
=> (1/ (sinx+1) ) + ( 1./sinx-1)
=> [(sinx-1) + (sinx +1) ]/ (sin² x -1)
=> [sinx+ sinx -1+1 ] / [sin² x - (sin² x+ cos² x) ]
=> [ 2sinx] / [ sin² x -sin² x -cos² x ]
=> 2sinx / -cos² x
Divide by negative 1 numerator and denominator
=> -2sinx / cos² x
=> Right hand side
.Q.teach that (cosecx+a million)^2/(cot^2x)=(cosecx+a million)/(cosecx... ? Ans : LHS : (cosecx+a million)^2/(cot^2x) RHS : {(cosecx+a million)/(cosecx-a million) } Multiply the Numerator and Denominator by utilising "(cosecx + a million)" => {(cosecx+a million)/(cosecx-a million) } * [(cosecx+a million)/(cosecx+a million) ] => { (cosecx+a million)*(cosecx+a million) } / [ (cosecx+a million)*(cosecx-a million) ] => { (cosecx+a million)^2 } / [ (cosecx+a million)*(cosecx-a million) ] note : (a+b)*(a-b) = a^2 - b^2 => { (cosecx+a million)^2 } / [ (cosec^2x- a million^2) ] => { (cosecx+a million)^2 } / [ (cosec^2x- a million) ] note : cosec^2y- a million = cot^2y therefore : {(cosecx+a million)^2 } / [(cot^2x)] =>(cosecx+a million)^2/(cot^2x) LHS = RHS (for this reason PROVED) I favor you'll rejoice with my answer.