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

Question

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

✓

Answer

Given,
$\sin\theta+\cos\theta=\text{x}$
Squaring the given equation, we have
$(\sin\theta+\cos\theta)^2=\text{x}^2$
$\Rightarrow\ \sin^2+2\sin\theta\cos\theta+\cos^2\theta=\text{x}^2$
$\Rightarrow\ (\sin^2\theta+\cos^2\theta)+2\sin\theta\cos\theta=\text{x}^2$
$\Rightarrow\ 1+2\sin\theta\cos\theta=\text{x}^2$
$\Rightarrow\ 2\sin\theta\cos\theta=\text{x}^2-1$
$\Rightarrow \sin\theta\cos\theta=\frac{\text{x}^2-1}{2}$
Squaring the last equation, we have
$(\sin\theta\cos\theta)^2=\frac{(\text{x}^2-1)^2}{4}$
$\Rightarrow \sin^2\theta\cos^2\theta=\frac{(\text{x}^2-1)^2}{4}$
Therefore, we have
$\sin^6\theta+\cos^6\theta=(\sin^2\theta)^3+(\cos^2\theta)^3$
$=(\sin^2\theta+\cos^2\theta)^3-3\sin^2\theta\cos^2\theta(\sin^2\theta+\cos^2\theta)$
$=(1)^3-3\frac{(\text{x}^2-1)^2}{4}(1)$
$=1-3\frac{(\text{x}^2-1)^2}{4}$
$=\frac{4-3(\text{x}^2-1)^2}{4}$
Hence 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 "4 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→Maharashtra Board STD 10 question papers→Maharashtra Board question paper generator→

Similar questions

Mr. Purushottam invested Rs. 1,20,354 in shares of face value Rs. 100 each at Rs.120 market value. He gave brokerage of 0.25% and GST of 18% on brokerage then how many shares are purchased by him?

Calculate the missing frequency from the following distribution, if the median of distribution is 24.

Age in years	0 – 10	10 – 20	20 – 30	30 – 40	40 – 50
No. of persons	5	25		18	7

In a four-sided field, the length of the longer diagonal is 128m. The lengths of perpendiculars from the opposite vertices upon this diagonal are 22.7m and 17.3m Find the area of the field.

Is it possible to design a rectangular mango grove whose length is twice its breadth and the area is $800m^2$?
If so, find its length and breadth.

From the top of a hill, the angles of depression of two consecutive kilometre stones due east are fouud to be 45° and 30° respectively.
Find the height of the hill.

Determine two consecutive multiples of $3$ whose product is $270.$

Find the values of a and b for which the following system of linear equations has infinite number of solutions:

$(a + b)x - (a + b - 3)y = 4a + b$

Construct a triangle of sides $4\ cm, 5\ cm$ and $6\ cm$ and then a triangle similar to it whose sides are $\Big(\frac{2}{3}\Big)$ of the corresponding sides of it.

The area of a rectangle is $192\ cm^2 $ and its perimeter is $56\ cm$. Find the dimensions of the rectangle.

Solve for $x$ and $y$:
$6(ax + by) = 3a + 2b,$
$6(bx - ay) = 3b - 2a$