In a , Prove that: A+B 2 = C 2 - Vidyadip

Question

In a $\triangle\text{ABC},$ Prove that: $\tan\frac{\text{A+B}}{2}=\cot\frac{\text{C}}{2}$

✓

Answer

We have: $\text{A + B + C} = \pi$ $\big(\because$ sum of 3 angle of a triangle is $\pi=180^\circ\big)$ $\Rightarrow\text{A+B}=\pi-\text{C}$ $\Rightarrow\frac{\text{A+B}}{2}=\frac{\pi-\text{C}}{2}$ $\Rightarrow\frac{\text{A+B}}{2}=\frac{\pi}{2}-\frac{\text{C}}{2}$ $\Rightarrow\tan\Big(\frac{\text{A+B}}{2}\Big)=\tan\Big(\frac{\pi}{2}-\frac{\text{C}}{2}\Big)$ $=\cot\frac{\text{C}}{2}$ $\Big(\because\tan\Big(\frac{\pi}{2}-\theta\Big)=\cot\theta\Big)$ Hence, $\tan\Big(\frac{\text{A+B}}{2}\Big)=\tan\Big(\frac{\pi}{2}-\frac{\text{C}}{2}\Big)$ $\text{Proved}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "(Each question 3 marks)" from Trigonometric Functions→Trigonometric Functions — all question types→MATHS STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The number of terms of an A.P. is even; the sum of odd terms is 24, of the even terms is 30, and the last term exceeds the first by $10\frac{1}{2},$ find the number of terms and the series.

$2(\text{bc}\cos\text{A}+\text{ca}\cos\text{B}+\text{ab}\cos\text{C})=\text{a}^2+\text{b}^2+\text{c}^2$

Find the values of the following expressions: $\text{i}+\text{i}^{2}+\text{i}^{3}+\text{i}^{4}$

$\frac{\text{b}\sec\text{B + c}\sec\text{C}}{\tan\text{B}+\tan\text{C}}=\frac{\text{c}\sec\text{C + a}\sec\text{A}}{\tan\text{C}+\tan\text{A}}=\frac{\text{a}\sec\text{A + b}\sec\text{B}}{\tan\text{A}+\tan\text{B}}.$

If $\text{f(x)}=\begin{cases}\text{x}^2,&\text{when }\text{ x}<0\\\text{x},&\text{when }\ 0\leq\text{x}<1\\\frac{1}{\text{x}},&\text{when }\text{ x}>0\end{cases}$ Find:

$\text{f}\Big(\frac{1}{2}\Big)$
$\text{f}(-2)$
$\text{f}(1)$
$\text{f}(\sqrt{3})$
$\text{f}(\sqrt{-3})$

Prove that: $\frac{\tan^22\text{x}-\tan^2\text{x}}{1-\tan^22\text{x}\tan^2\text{x}}=\tan3\text{x}\tan\text{x}$

Find $\text{f}+\text{g},\text{ f}-\text{g},\text{ cf}(\text{c}\in\text{ R},\text{c}\neq0),\text{ fg},\frac{1}{\text{f}}$ and $\frac{\text{f}}{\text{g}}$ in the following: If $f(x) = x^3 + 1$ and $g(x) = x + 1$

A fair coin is tossed four times, and a person win Re $1$ for each head, and lose Rs. $1.50$ for each tail that turns up. Form the sample space, calculate how many different amounts of money he can have after four tosses and the probability of having each of these amounts.

Evaluate the following: $^{n+1}C_n$

If $\cot \text{x}(1+\sin\text{x})=4\text{ m}$ and $\cot \text{x}(1-\sin\text{x})=4\text{ n},$prove that $\text{(m}^2-\text{n}^2)^2=\text{mn.}$