In ABC, C=2 3 , then prove that ^2 ~A+ ^2 ~B- A B=3 4 . - Vidyadip

Question

In $\triangle \mathrm{ABC}, \angle \mathrm{C}=\frac{2 \pi}{3}$, then prove that $\cos ^2 \mathrm{~A}+\cos ^2 \mathrm{~B}-\cos \mathrm{A} \cos \mathrm{B}=\frac{3}{4}$.

✓

Answer

$\begin{aligned}
& \angle C=\frac{2 \pi}{3} \\
& \text { In } \triangle \mathrm{ABC} \text {, } \\
& \mathrm{A}+\mathrm{B}+\mathrm{C}=\pi \\
& \therefore \quad \mathrm{A}+\mathrm{B}+\frac{2 \pi}{3}=\pi \\
& \therefore \quad \mathrm{A}+\mathrm{B}=\frac{\pi}{3} \\
& \therefore \quad \cos (A+B)=\cos \frac{\pi}{3}=\frac{1}{2} \\
& \text { L.H.S. }=\cos ^2 \mathrm{~A}+\cos ^2 \mathrm{~B}-\cos \mathrm{A} \cos \mathrm{B} \\
& =\frac{1}{2}\left(2 \cos ^2 A+2 \cos ^2 B\right)-\frac{1}{2}(2 \cos A \cos B) \\
& =\frac{1}{2}(1+\cos 2 \mathrm{~A}+1+\cos 2 \mathrm{~B}) \\
& -\frac{1}{2}(2 \cos A \cos B) \\
& =\frac{1}{2}\left[2+2 \cos \left(\frac{2 \mathrm{~A}+2 \mathrm{~B}}{2}\right) \cos \left(\frac{2 \mathrm{~A}-2 \mathrm{~B}}{2}\right)\right] \\
& -\frac{1}{2}[\cos (A+B)+\cos (A-B)] \\
& =\frac{1}{2}[2+2 \cos (A+B) \cos (A-B)] \\
& -\frac{1}{2}[\cos (A+B)+\cos (A-B)] \\
& =\frac{1}{2}\left[2+2\left(\frac{1}{2}\right) \cos (A-B)\right] \\
& -\frac{1}{2}\left[\frac{1}{2}+\cos (A-B)\right] \\
& \text {...[From (i)] } \\
& =\frac{1}{2}\left[2+\cos (A-B)-\frac{1}{2}-\cos (A-B)\right] \\
& =\frac{1}{2}\left(\frac{3}{2}\right)=\frac{3}{4}\\
& =\text { R.H.S. } \\
\end{aligned}$

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 "Solve the Following Question.(5 Marks)" from Trigonometry – II→Trigonometry – II — all question types→Maths STD 11 Science question papers→Maharashtra Board STD 11 Science question papers→Maharashtra Board question paper generator→

Similar questions

Find the equations of the circles touching $y-$axis at $(0, 3)$ and making an intercept of $8$ units on the $x-$axis.

Find the equation of the hyperbola whose,
Focus is (1, 1) directrix is $\text{2x}+\text{y}=1$ and eccentricity $= \sqrt{3}$

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow1}\bigg\{\frac{\text{x}-2}{\text{x}^2-\text{x}}-\frac{1}{\text{x}^2-3\text{x}^2+2\text{x}}\bigg\}$

In each the following find the equation of the hyperbola satisfying the given conditions:
Foci $(0, \pm13), $ conjugate axis = 24

Prove the following by the principle of mathematical induction:
$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2^\text{n}}=1-\frac{1}{2^\text{n}}$

If $a, b, c$ are in G.P., prove that:
$\frac{\text{a}^2+\text{ab}+\text{b}^2}{\text{bc}+\text{ca}+\text{ab}}=\frac{\text{b}+\text{a}}{\text{c}+\text{b}}$

Represent to solution set of each of the following inequations graphically in two dimensional plane:
$\text{x}+2\geq0$

Prove the following by the principle of mathematical induction:
$\frac{1}{2.5}+\frac{1}{5.8}+\frac{1}{8.11}+...+\frac{1}{(\text{3n-1)(3n+2)}}=\frac{\text{n}}{\text{6n}+4}$

Sum the following series to $n$ terms:
$2 + 4 + 7 + 11 + 16 + ......$

Find the co-ordinates of the circumcentre of the triangle whose vertices are A(- 2, 3), B(6, -1), C(4,3),e