Prove that: ^22 5 = ^2 3 = 5-1 8 - Vidyadip

Question

Prove that:
$\sin^2\frac{2\pi}{5}=\sin^2\frac{\pi}{3}=\frac{\sqrt{5}-1}{8}$

✓

Answer

We have,
$\sin^272^\circ-\sin^260^\circ,$
$=\sin^2(90^\circ-18^\circ)-\Big(\frac{\sqrt{3}}{2}\Big)^2$
$=\cos^218^\circ-\frac{3}{4}$
$\Bigg(\frac{\sqrt{10+2+\sqrt{5}}}{4}\Bigg)^2-\frac{3}{4}\Bigg[\because\cos18^\circ=\frac{\sqrt{10+2\sqrt{5}}}{4}\Bigg]$
$=\frac{10+2\sqrt{5}}{16}-\frac{3}{4}$
$=\frac{10+2\sqrt{5}-12}{16}$
$=\frac{2\sqrt{5}-2}{16}$
$=\frac{\sqrt{5}-1}{8}$

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 "2 Marks Questions" from Trigonometric Ratios Of Compound Angles→Trigonometric Ratios Of Compound Angles — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Find the probability that in a random arrangement of the letters of the word 'UNIVERSITY', the two I's do not come together.

How many different products can be obtained by multiplying two or more of the numbers 3, 5, 7, 11 (without repetition)?

Show the following quadratic equation by factorization method:
x²+ (1 − 2i) x − 2i = 0

In a simultanepous throw of a pair of dice, find the probability of getting,

A sum less than 6.

Find the indicated terms of the sequence, whose nth term is $a _ { n } = \frac { n ( n - 2 ) } { n + 3 }$; a₂₀

If $\text{P}= \big\{\text{x} : \text{x} < 3, \text{x}\in\text{N}\big\},$ $\text{Q}= \big\{\text{x} : \text{x} \leq 2, \text{x}\in\text{W}\big\}.$ Find $(\text{P} \cup \text{Q}) \times (\text{P} \cap \text{Q}),$ where W is the set of whole numbers.

Find the equation of hyperbola, when foci are at ($\pm$5, 0) and transverse axis is of length 8.

Calculate the value of $\lim _{x \rightarrow a} \frac{a \sin x-x \sin a}{a x^2-a^2 x}.$

Find the distance of the point (-1, 1) from the line 12 (x + 6) = 5(y - 2).

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow{\frac{\pi}{2}}}\Big(\frac{\pi}{2}-\text{x}\Big)\tan\text{x}$