Question

Prove that tan 44° x tan 46° = 1.

Answer 1

Recall the angle sum identities for cosine,

cos(x + y) = cos(x) cos(y) - sin(x) sin(y)

cos(x - y) = cos(x) cos(y) + sin(x) sin(y)

Now, by definition of tangent, we have

tan(x) = sin(x) / cos(x)

⇒ tan(44°) tan(46°) = sin(44°) sin(46°) / (cos(44°) cos(46°))

From the identities above, we can show that

cos(x) cos(y) = 1/2 (cos(x - y) + cos(x + y))

sin(x) sin(y) = 1/2 (cos(x - y) - cos(x + y))

so that

sin(44°) sin(46°) = 1/2 (cos(44° - 46°) - cos(44° + 46°))

⇒ sin(44°) sin(46°) = 1/2 (cos(-2°) - cos(90°))

⇒ sin(44°) sin(46°) = 1/2 cos(2°)

and

cos(44°) cos(46°) = 1/2 (cos(44° - 46°) + cos(44° + 46°))

⇒ cos(44°) cos(46°) = 1/2 (cos(-2°) + cos(90°))

⇒ cos(44°) cos(46°) = 1/2 cos(2°)

and in turn, it follows that

tan(44°) tan(46°) = (1/2 cos(2°)) / (1/2 cos(2°))

⇒ tan(44°) tan(46°) = 1

Answer 2

We have to prove that tan (44°) tan (46°) = 1 . So , for this we will use the sum identity for the tangent function i.e.

• 
  `{\boxed{\bf{\tan (x+y)=(\tan (x)+\tan (y))/(1-\tan (x)\tan (y))}}}`

So , here , we can easily get the answer by just putting x = 44° and y = 46° but let's first re-write it and then simplify to get the term tan(x) tan (y) , so that the answer will be just equal to what the product of tan (x) and tan (y) equal to

`{:\implies \quad \sf \tan (x+y)\{1-\tan (x)\tan (y)\}=\tan (x)+\tan (y)}`

`{:\implies \quad \sf 1-\tan (x)\tan (y)=(\tan (x)+\tan (y))/(\tan (x+y))}`

`{:\implies \quad \sf \tan (x)\tan (y)=(\tan (x)+\tan (y))/(\tan (x+y))+1}`

Now , put x = 44° and y = 46°

`{:\implies \quad \sf \tan (44^(\circ))\tan (46^(\circ))=(\tan (44^(\circ))+\tan (46^(\circ)))/(\tan (44^(\circ)+46^(\circ)))+1}`

`{:\implies \quad \sf \tan (44^(\circ))\tan (46^(\circ))=(\tan (44^(\circ))+\tan (46^(\circ)))/(\tan (90^(\circ)))+1}`

`{:\implies \quad \sf \tan (44^(\circ))\tan (46^(\circ))=(\tan (44^(\circ))+\tan (46^(\circ)))/(\infty)+1\quad \qquad \{\because \tan (90^(\circ))=\infty\}}`

`{:\implies \quad \sf \tan (44^(\circ))\tan (46^(\circ))=0+1}`

`{:\implies \quad \bf \therefore \quad \underline{\underline{\tan (44^(\circ))\tan (46^(\circ))=1}}}`

Hence , Proved