Question

prove that : sina cosa (tana + cota)=1​

Answer 1

$\underline{\underline{\sf{\clubsuit \:\:Question}}}$

$\bullet\:\:\tt{\mathrm{Prove\:}\sin \left(a\right)\cos \left(a\right)\left(\tan \left(a\right)+\cot \left(a\right)\right)=1}$

$\underline{\underline{\sf{\clubsuit \:\:Answer}}}$

$\tt{\sin \left(a\right)\cos \left(a\right)\left(\tan \left(a\right)+\cot \left(a\right)\right)=1}$

$\sf{Manipulating\:left\:side}$

$\tt{\sin \left(a\right)\cos \left(a\right)\left(\tan \left(a\right)+\cot \left(a\right)\right)}$

$\tt=\left(\dfrac{\cos \left(a\right)}{\sin \left(a\right)}+\tan \left(a\right)\right)\cos \left(a\right)\sin \left(a\right)}$

$\tt{=\left(\dfrac{\cos \left(a\right)}{\sin \left(a\right)}+\dfrac{\sin \left(a\right)}{\cos \left(a\right)}\right)\cos \left(a\right)\sin \left(a\right)}$

$\tt{=\dfrac{\cos ^2\left(a\right)+\sin ^2\left(a\right)}{\sin \left(a\right)\cos \left(a\right)}\cos \left(a\right)\sin \left(a\right)}$

$\tt{=\dfrac{\left(\cos ^2\left(a\right)+\sin ^2\left(a\right)\right)\cos \left(a\right)\sin \left(a\right)}{\sin \left(a\right)\cos \left(a\right)}}$

$\tt{=\dfrac{\left(\cos ^2\left(a\right)+\sin ^2\left(a\right)\right)\sin \left(a\right)}{\sin \left(a\right)}}$

$\tt{=\cos ^2\left(a\right)+\sin ^2\left(a\right)}$

$\sf{Use\:the\:following\:identity:\quad \cos ^2\left(x\right)+\sin ^2\left(x\right)=1}$

$\tt{=1}$

$\Large\boxed{\sf{{L.H.S\:=\:R.H.S}}}$

Hence Proved !!!