verify tan (sin+cot×cos)=sec

Question

Verify the identity

Answer 1

tan (sin+cot×cos)=sec

 

 

LHS = tanx(sinx + cotxcosx)

            =tanxsinx + tanxcotxcosx

           =(sinx/cosx)sinx + 1*cosx         [ tanx = sinx/cosx;   tanxcotx =1]

          = (sin^2x/cosx )+ cosx

         =( sin^2x +cos^2x)/cosx                 [ LCM = cosx ]

         = 1/cosx                           [ sin^2x + cos^2x = 1]

        = secx                       [ i/cosx = secx]

        = LHS

 

Answer 2

given tan(sin+cot x cos)

=tan(sin+(1/tan)xcos)

=tan(sin+(1/sin/cos)xcos)

=tan(sin+(cos/sin)xcos)

=tan(sin+cos^2/sin

=tan(sin^2+cos^2)/sin

=tan/sin

=(sin/cos)/sin

=1/cos

=sec

Answer 3

tan(sin+cot×cos)=sec

From RHS tanx{(sinx+cotxcosx)}

=tanx{sinx+[cosx/sinx]cosx}

=tanx{sinx+cos^2(x)/sinx}

=tanx{[sin^2(x)+cos^2(x)]/sinx} ,   sin^2(x)+cos^2(x)=1

=tanx[1/sinx]

=tanx/sinx

=1/cosx

=secx Verified

Therefore, it is an identity