Download App

Trigonometric Functions — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsTrigonometric Functions3 Marks
Question
Prove the following identities:
$\sin^6\text{x}+\cos^6\text{x}=1−3\sin^2\text{x}\cos^2\text{x}$
✓

Answer

$\text{L.H.S}=\sin^{6}\text{x}+\cos^{6}\text{x}$
$=\big(\sin^{2}\text{x}\big)^{3}+\big(\cos^{6}\text{x}\big)$
$=\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)$ $\bigg[\big(\sin^{2}\text{x}\big)^{2}-\big(\sin^{2}\text{x}\big)-\sin^{2}\text{x}\cos^{2}\text{x}+\big(\cos^{2}\text{x}\big)\bigg]$ $\big(\because\text{a}^{3}+\text{b}^{3}=\big(\text{a}+\text{b}\big)\big(\text{a}^{2}-\text{ab}+\text{b}^{2}\big)$
$=\big(\sin\text{x}\big)^{2}+\big(\cos\text{x}\big)^{2}$ $\Big[\big(\sin^{2}\text{x}\big)^{2}+2\sin^{2}\text{x}\cos^{2}\text{x}-2\sin^{2}\text{x}\cos^{2}\text{x}-\sin^{2}\cos^{2}\text{x}\big)\Big]$
$=\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)^{2}-3\sin^{2}\text{x}\cos^{2}\text{x}$
$=\big(\sin^{2}\text{x}+\cos^{2}\text{x}\big)^{2}-3\sin^{2}\text{x}\cos^{2}\text{x}$
$=1^{2}-3\sin^{2}\text{x}\cos^{2}\text{x}$ $\big(\because\sin^{2}\text{x}+\cos^{2}\text{x}=1\big)$
$=1-3\sin^{2}\text{x}\cos^{2}\text{x}$
$=\text{R.H.S}$
$\text{L.H.S = R.H.S}$

Need a full question paper?

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

Start Generating Free

Explore more

More "Solve the Following Question.(3 Marks)" from Trigonometric Functions→Trigonometric Functions — 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 equation of the right bisector of the line segment joining the points A(1, 0) and B(2, 3).
→
Find the equation of the line which passes through the point of intersection of lines $x+2 y+6=0, \quad 2 x-y=2$ and which makes intercept 5 on the Y-axis.
→
Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sec5\text{x}-\sec3\text{x}}{\sec3\text{x}-\sec\text{x}}$
→
Evaluate the following limits: $\lim _{x \rightarrow 0}\left[\frac{a^x-b^x}{\sin (4 x)-\sin (2 x)}\right]$
→
Prove that:
$\cot^2\text{x}-\tan^2\text{x}=4\cot2\text{x}\ \text{cosec}\ 2\text{x}$
→
Find the equation of the straight line on which the length of the perpendicular from the origin makes an angle of 30° with x-axis and which forms a triangle of area $\frac{50}{\sqrt3}$ with the axes.
→
$1+\text{i}^2+\text{i}^4+\text{i}^6+\text{i}^8+ ...+\text{i}^{20}$
→
Let A = {1, 2, 3, 4} and $\text{R}=\{(\text{a},\text{b}):\text{a}\in\text{A},\text{b}\in\text{A},\text{ a divides b}\}.$ Write R explicitly.
→
Find the equation of the straight lines passing through the following pair of points:
(0, -a) and (b, 0)
→
Let f(x) be a function defined by $\text{f(x)}=\begin{cases}\frac{3\text{x}}{|\text{x}|+2\text{x}}, & \text{x} \neq0\\\ \ \ \ 0, & \text{x} = 0\end{cases}.$
→