Question
Prove the following identities:
$\frac{\tan\text{A}+\tan\text{B}}{\cot\text{A}+\cot\text{B}}=\tan\text{A}\tan\text{B}$
$\frac{\tan\text{A}+\tan\text{B}}{\cot\text{A}+\cot\text{B}}=\tan\text{A}\tan\text{B}$
✓
Answer
$\text{LHS}=\frac{\tan\text{A}+\tan\text{B}}{\cot\text{A}+\cot\text{B}}$
$=\frac{\frac{\sin\text{A}}{\cos\text{A}}+\frac{\sin\text{B}}{\cos\text{B}}}{\frac{\cos\text{A}}{\sin\text{A}}+\frac{\cos\text{B}}{\sin\text{ B}}}$
$=\frac{\frac{\sin\text{B}\cos\text{B}+\sin\text{B}\cos\text{A}}{\cos\text{A}\cos\text{B}}}{\frac{\cos\text{A}\sin\text{B}+\cos\text{B}\sin\text{A}}{\sin\text{A}\sin\text{B}}}$
$=\frac{(\sin\text{A}\cos\text{B}+\sin\text{B}\cos\text{A})\times\sin\text{A}\sin\text{B}}{\cos\text{A}\cos\text{B}\times\big(\cos\text{A}\sin\text{B}+\cos\text{B}\sin\text{A}\big)}$
$=\frac{\sin\text{A}\sin\text{B}}{\cos\text{A}\cos\text{B}}$
$=\tan\text{A}\tan\text{B}$
$=\text{R.H.S}$
$\therefore\ \text{L.H.S.}=\text{R.H.S.}$
$=\frac{\frac{\sin\text{A}}{\cos\text{A}}+\frac{\sin\text{B}}{\cos\text{B}}}{\frac{\cos\text{A}}{\sin\text{A}}+\frac{\cos\text{B}}{\sin\text{ B}}}$
$=\frac{\frac{\sin\text{B}\cos\text{B}+\sin\text{B}\cos\text{A}}{\cos\text{A}\cos\text{B}}}{\frac{\cos\text{A}\sin\text{B}+\cos\text{B}\sin\text{A}}{\sin\text{A}\sin\text{B}}}$
$=\frac{(\sin\text{A}\cos\text{B}+\sin\text{B}\cos\text{A})\times\sin\text{A}\sin\text{B}}{\cos\text{A}\cos\text{B}\times\big(\cos\text{A}\sin\text{B}+\cos\text{B}\sin\text{A}\big)}$
$=\frac{\sin\text{A}\sin\text{B}}{\cos\text{A}\cos\text{B}}$
$=\tan\text{A}\tan\text{B}$
$=\text{R.H.S}$
$\therefore\ \text{L.H.S.}=\text{R.H.S.}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
If the sum of first m terms of an AP is $(2m^2+ 3m)$ then what is its second term?
→Find the area of quad. ABCD in which AB = 42cm, BC = 21cm, CD = 29cm, DA = 34cm and diag. BD = 20cm.
Solve for x and y:
217x + 131y = 913,
131x + 217y = 827
217x + 131y = 913,
131x + 217y = 827
The adjacent sides of a parallelogram ABCD measure 34cm and 20cm, and the diagonal AC measures 42cm. Find the area of the parallelogram.
If k, (2k - 1) and (2k + 1) are the three successive term of an AP, find the value of k.
The difference between the sides at right angle in a right-angled triangle is $7\ cm$. The area of the triangle is $60\ cm^2$. Find its perimeter.
→Find the area of the triangle whose sides are 42cm, 34cm and 20cm in length. Find the height corresponding to the longest side.
Two APs have the same common difference. If the first term of these APs be $3$ and $8$ respectively, find the difference between the sums of their first $50$ terms.
→If 2 is added to each of two given numbers, their ratio becomes 1 : 2. However, if 4 is subtracted from each of the given numbers, the ratio becomes 5 : 11. Find the numbers.
Show graphically that each of the following given systems of equations is inconsistent, i.e., has no solutions:
2x + y = 6, 6x + 3y = 20
2x + y = 6, 6x + 3y = 20