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 "5 Marks Questions" from Trigonometric Identities→Trigonometric Identities — all question types→Maths STD 10 question papers→CBSE Board STD 10 question papers→CBSE Board question paper generator→

Similar questions

The radii of the ends of a bucket 30cm high are 21cm and 7cm. Find its capacity in litres and the amount of sheet required to make this bucket.

On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs. 2000. But if he sells the T.V. at 10% gain the fridge at 5% loss. He gains Rs. 1500 on the transaction. Find the actual prices of T.V. and fridge.

Show that the points A(2, 1), B(5, 2), C(6, 4) and D(3, 3) are the angular points of a parallelogram. Is this figure a rectangle?

A metallic cylinder has radius 3cm and height 5cm. To reuce its weight, a conical hole is drilled in the cylinder. The conical hole has a radius of $\frac{3}{2}\text{cm}$ and its depth is $\frac{8}{9}\text{cm}.$ Calculate the ratio of the volume of metal left in the cylinder to the volume of metal taken out in conical shape.

A solid is in the shape of a cone surmounted on a hemisphere with both their diameters being equal to $7 \ cm$ and the height of the cone is equal to its radius. Find the volume of the solid.

Susan invested certain amount of money in two schemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively. She received Rs. 1860 as annual interest. However, had she interchanged the amount of investments in the two schemes, she would have received Rs. 20 more as annual interest. How much money did she invest in each scheme?

Calculate the mode from the following data:

Monthly salary (in Rs)	No. of employees
0-5000	90
5000-1000	150
10000-15000	100
15000-20000	80
20000-25000	70
25000-30000	10

Find the area of a quadrant of a circle whose circumference is 88cm. $[\text{Take }\pi\ =3.14]$

Prove that the opposite sides of a quadrilateral circumscribing a circle subtend supplementary angles at the centre of the circle.

Solve the following systems of equations graphically:

2x - 3y + 8 = 0