Question

Prove that (sinx+secx)2 +(cosx+cosec)2 = (1+secx- cosecx)2

Answer 1

Correct question :-

Prove that $\sf (sinx+secx)^2+(cosx+cosecx)^2 = (1+secx \ cosecx )^2$ .

Solution :-

$\sf L.H.S = (sinx+secx)^2+(cosx+cosecx)^2$

$\bullet \bf \ secx= \dfrac{1}{cosx} \\\\\bullet \ cosec x = \dfrac{1}{sinx}$

      $= \sf \left( sinx +\dfrac{1}{cosx} \right)^2 + \left(cos x +\dfrac{1}{sinx} \right)^2 \\\\= \left( \dfrac{sinxcosx+1} {cosx} \right) ^2 + \left( \dfrac{sinxcosx+1}{sinx} \right)^2 \\\\= (sinxcosx+1)^2 \times \left[ \dfrac{1}{cos^2x}+\dfrac{1}{sin^2x} \right] \\\\= (sinxcosx+1)^2 \times \dfrac{sin^2x +cos^2x }{sin^2xcos^2x}$

$\bullet \bf \ sin^2x+cos^2x = 1$

       $= \sf (sinxcosx+1)^2 \times \dfrac{1}{sin^2xcos^2x} \\\\= \left(\dfrac{sinxcosx+1}{sinxcos}\right)^2 \\\\= \left( \dfrac{sinxcosx}{sinxcosx} ^2 +\dfrac{1}{sinxcosx} \right)^2 \\\\= \left( 1+\dfrac{1}{sinxcosx} \right) ^2 \\\\= \left(1 + \dfrac{1}{sinx} \times \dfrac{1}{cosx} \right)^2 \\\\= (1+cosecx \ secx )^2 \\\\= (1+secx \ cosecx) ^2\\\\= R.H.S$

Hence proved.

Answer 2

$\underline{\sf{\red{Correct \: Question:-}}}$

$\sf\ (sin x + sec x)^2 + ( cos x + cosec x)^2$

$\sf\ = ( 1+secx-cosec x)^2$

$\underline{\sf{\red{Given:-}}}$

• $\sf\ (sin x + sec x)^2 + ( cos x + cosecx)^2$

$\sf\ = ( 1+secx-cosec x)^2$

$\underline{\sf{\red{Solution:-}}}$

$\sf\ LHS$

$\longrightarrow\: \sf\ (sinx+secx)^2 + (cos x + cosec x)^2$

$\longrightarrow\: \sf\ (sin x + \dfrac{1}{cos x} )^2 + (cos x + \dfrac{1}{sin x} )^2$

$\longrightarrow\: \sf\ sin^2x + \dfrac{1}{cos^2x} + 2 \dfrac{sin x}{cos x} + cos^2x +$

$\sf\ \dfrac{1}{sin^2x} + 2 \dfrac{cos x}{sin x}$

$\longrightarrow\: \sf\ (sin^2x + cos^2x) + ( \dfrac{1}{cos^2x} + \dfrac{1}{sin^2x} )$

$\sf\ + 2 (\dfrac{ sin x}{cos x} + \dfrac{cos x}{sin x}$

$\longrightarrow\: \sf\ 1 + \dfrac{1}{sin^2x . cos^2 x} + \dfrac{2}{sin x . cos x}$

$\longrightarrow\: \sf\ \dfrac{ sin^2x . cos^2 + 1+2 sin x . cos x}{sin^2x . cos^2x}$

$\longrightarrow\: \sf\ ( \dfrac{sin x. cos x +1}{sin x. cos x} )^2$

$\longrightarrow\: \sf\ (1 + sec x. cosec x^2)^2$

Hence,

• LHS = RHS