Question

Show that RHS=LHS
Std X SSC Board Maharashtra

Answer 1

I m writing "x" in place of theta
L.H.S=tanx/(1-cotx) + cotx/(1-tanx)

=(sinx÷cosx)/{1-(cosx÷sinx)} + (cosx÷sinx)/{1-(sinx/cosx)}

=(sinx÷cosx)/{(sinx-cosx)÷sinx} + (cosx÷sinx)/{(cosx-sinx)÷cosx}

=(sinx÷cosx) × {sinx÷(sinx-cosx)} + (cosx÷sinx) × {cosx÷(cosx-sinx)}

=sin^2x/cosx(sinx-cosx) + cos^2x/sinx(cosx-sinx)

=sin^2 x/cosx(sinx-cosx) - cos^2 x/sinx(sinx-cosx)
(Changing the symbol between the 2 terms to change the sign in the denominator)

={(sin^2 x × sinx) - (cos^2 x × cosx)} /{sinx.cosx.(sinx-cosx)}

=(sin^3 x-cos^3 x)/{sinx.cosx.(sinx-cosx)}

={(sinx-cosx)(sin^2 x + sinx.cosx +cos^2 x)}/{sinx.cosx.(sinx-cosx)
......{a^3 -b^3 = (a-b)(a^2+ab+b^2)}, well this is an important formula if you dont know do learn it

=(sin^2 x+cos^2 x+sinx.cosx)/(sinx.cosx)

=(1+sinx.cosx)/(sinx.cosx)
......(sin^2 A+ cos^2 A=1)

={1+(1÷cosecx.secx)}/ (1÷cosecx.secx)

={(cosecx.secx+1)÷cosecx.secx}/ (1÷cosecx.secx)

=(cosecx.secx+1)/cosecx.secx × cosecx.secx/1

=cosecx.secx + 1

=1+cosecx.secx=R.H.S.

Hence proved.

Answer 2

Step-by-step explanation:

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