If + =m, then prove that ^6x+ ^6x= 4-3(m^2-1)^2 4 , where m^2 2 - Vidyadip

Question

If $\sin\text{x}+\sin\cos\text{x}=\text{m},$ then prove that $\sin^6\text{x}+\cos^6\text{x}=\frac{4-3(\text{m}^2-1)^2}{4},$ where $\text{m}^2\leq2$

✓

Answer

To show: $\sin^6\text{x}+\cos^6\text{x}=\frac{4-3(\text{m}^2-1)^2}{4},$ $\text{ where m}^2\leq2$ Since, $\sin\text{x}+\cos\text{x}=\text{m}\cdots(\text{i})$ $\Rightarrow(\sin\text{x}+\cos\text{x})^2=\text{m}^2$ $\Rightarrow\sin^2\text{x}+\cos^2\text{x}+2\sin\text{x}\cos\text{x}=\text{m}^2$ $\Rightarrow1+2\sin\text{x}\cos\text{x}=\text{m}^2$ $(\because\sin^2\text{x}+\cos^2\text{x}=1)$ $\Rightarrow2\sin\text{x}\cos\text{x}=\text{m}^2-1$ $\Rightarrow\sin\text{x}\cos\text{x}=\frac{\text{m}^2-1}{2}\cdots(\text{ii})$ $\therefore\text{L.H.S}=\sin^6\text{x}+\cos^2\text{x}$ $=(\sin^2\text{x})^3+(\cos^3\text{x})^3$ $=(\sin^2\text{x}+\cos^2\text{x})(\sin^2\text{x})^2+(\cos^2\text{x})^2-\sin^2\text{x}\cos^2\text{x}$ $=1.((\sin^2\text{x})^2+(\cos^2\text{x})^2\\\ \ \ +2\sin^2\text{x}\cos^2\text{x}-2\sin^2\text{x}\cos^2\text{x}-\sin^2\text{x}\cos^2\text{x})$ $(\text{adding and subtracting }2\sin^2\text{x}\cos^2\text{x})$ $=(\sin^2\text{x}+\cos^2\text{x})^2-3\sin^2\text{x}\cos^2\text{x}$ $=1-3\sin^2\text{x}\cos^2\text{x}$ $=1-3(\sin\text{x}\cos\text{x})^2$ $=1-3\frac{(\text{m}^2-1)^2}{4}$ $\text{(from (ii))}$ $=\frac{4-3(\text{m}^2-1)^2}{4}, \text{ where m}^2\leq2$ $=\text{R.H.S}$ $\text{Proved}$

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

Find the area of the triangle formed by the lines joining the vertex of the parabola $x^2 = 12y$ to the ends of its latus rectum.

Find the equation to the ellipse in the following case: eccentricity $\text{e}=\frac{1}{2}$ and major axis = 12

Prove that: $\cos^32\text{x}+3\cos2\text{x}=4(\cos^6\text{x}-\sin^6\text{x})$

What is the number of ways of choosing 4 cards from a pack of 52 playing cards? In how many of these four cards belong to four different suits.

The perpendicular from the origin to the line y = mx + c meets it at the point (-1, 2). Find the values of m and c.

Prove the following by using the principle of mathematical induction for all n ∈ N:$1+2+3+...+\text{n}<\frac{1}{8}(2\text{n}+1)^2.$

The number of telephone calls received at an exchange in 245 successive one- minute intervals are shown in the following frequency distribution:

Number of calls	0	1	2	3	4	5	6	7
Frequency	14	21	25	43	51	40	39	12

Compute the mean deviation about median.

Prove the following by using the principle of mathematical induction for all n ∈ N:$1.2+2.3+3.4+...+\text{n.(n+1)}=\Big[\frac{\text{n(n+1)(n+2)}}{3}\Big].$

If A = {-1, 1}, find A × A × A.

In $\triangle\text{ABC}$ prove that, it $\theta$ be any angle, then $\text{b}\cos\theta=\text{c}\cos(\text{A}-\theta)+\text{a}\cos(\text{C}+\theta).$