Download App

Trigonometric Functions — MATHS STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 ScienceMATHSTrigonometric Functions4 Marks
Question
If $\text{T}_\text{n}=\sin^\text{n}\text{x}+\cos^\text{n}\text{x},$ Prove that $2 \text{T}_6 - 3\text{ T}_4 + 1 = 0$
✓

Answer

$\text{L.H.S}=2\text{T}_6-3\text{T}_4+1$
$=2(\sin^6\text{x}+\cos^6\text{x})-3(\sin^4\text{x}+\cos^4\text{x})+1$
$=2((\sin^2\text{x})^3+(\cos^2\text{x})^3-2(\sin^2\text{x})^2+(\cos^2\text{x})^2)+1$
$=2((\sin^2\text{x}+\cos^2\text{x})(\sin^2\text{x})^2+(\cos^2\text{x})^2-(\sin^2\text{x}\cos^2\text{x}))$
$\ \ \ -3((\sin^2\text{x})^2+(\cos^2\text{x})^2+2\sin^2\text{x}\cos^2\text{x}-2\sin^2\text{x}\cos^2\text{x})=1$
$ [$Using $a^3 + b^3 =(a + b)(a^2 + b^2 - ab)$ and adding and subtracting $2\sin^2\text{x}\cos^2\text{x}\big]$
$=2((\sin^2\text{x}+\cos^2\text{x})^2-3\sin^2\text{x}\cos^2\text{x})-3(1-2\sin^2\text{x}\cos^2\text{x})+1$
$=2(1-3\sin^2\text{x}\cos^2\text{x})-3\sin^2\text{x}\cos^2\text{x}+1$
$=2-6\sin^2\text{x}\cos^2\text{x}-2+6\sin^2\text{x}\cos^2\text{x}$
$=0$
$=\text{R.H.S}$
$\text{Proved}.$

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 4 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

Differentiate the following functions with respect to x:$\frac{1+3^\text{x}}{1-3^\text{x}}$
→
Solve the following equation: $\sqrt{3}\text{x}^2-\sqrt{2}\text{x}+3\sqrt{3}=0.$
→
Find the equations to the straight lines which go through the origin and trisect the portion of the straight line 3x + y = 12 which is intercepted between the axes of coordinates.
→
Find the point of intersection of the following pairs of lines: $\text{y}=\text{m}_1\text{x}+\frac{\text{a}}{\text{m}_1}$ and $\text{y}=\text{m}_2\text{x}+\frac{\text{a}}{\text{m}_2}$
→
Show that the product of perpendiculars on the line $\frac{\text{x}}{\text{a}}\cos\theta+\frac{\text{y}}{\text{b}}\sin\theta=1$ from the points $\Big(\sqrt{\text{a}^2-\text{b}^2},0\Big)$ is $\text{b}^2.$
→
Find the derivative of 99x at x = l00 .
→
Find the equation of the hyperbola whoseFoci are (6, 4) and (-4, 4) and eccentricity is 2.
→
If A(-1, 1) and B(2, 3) are two fixed points, find the locus of a point P, so that the area of $\triangle\text{PAB} = 8$ sq. units.
→
For any two sets A and B, prove that: $\text{A}\cap\text{B}=\phi\Rightarrow\text{A}\subseteq\text{B}'.$
→
Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers): $(\text{x}+\cos\text{x})(\text{x}-\tan\text{x})$
→