sin7 θ= (a) sin^7 θ-35 sin^5 θcos^2 θ+21 sin^3 θcos^4 θ-cos^7 θ(b) sin^6 θcosθ-35 sin^5 θcos^2 θ

step 1 Here in this problem we have to find the value of sine 7 theta by choosing the appropriate optio

Sin7 θ= (a) sin^7 θ-35 sin^5 θcos^2 θ+21 sin^3 θcos^4...

$\sin 7 \theta=$ (a) $\sin ^{7} \theta-35 \sin ^{5} \theta \cos ^{2} \theta+21 \sin ^{3} \theta \cos ^{4} \theta-\cos ^{7} \theta$ (b) $\sin ^{6} \theta \cos \theta-35 \sin ^{5} \theta \cos ^{2} \theta+21 \sin ^{4} \theta \cos ^{3} \theta-\cos ^{7} \theta$ (c) $7 \cos ^{7} \theta-35 \cos ^{4} \theta \sin ^{3} \theta+21 \cos ^{2} \theta \sin ^{5} \theta+\sin ^{7} \theta$ (d) $(\sin \theta)\left[7 \cos ^{6} \theta-35 \cos ^{4} \theta \sin ^{2} \theta+21 \cos ^{2} \theta \sin ^{4} \theta-\sin ^{6} \theta\right]$

$\sin 7 \theta=$
(a) $\sin ^{7} \theta-35 \sin ^{5} \theta \cos ^{2} \theta+21 \sin ^{3} \theta \cos ^{4} \theta-\cos ^{7} \theta$
(b) $\sin ^{6} \theta \cos \theta-35 \sin ^{5} \theta \cos ^{2} \theta+21 \sin ^{4} \theta \cos ^{3} \theta-\cos ^{7} \theta$
(c) $7 \cos ^{7} \theta-35 \cos ^{4} \theta \sin ^{3} \theta+21 \cos ^{2} \theta \sin ^{5} \theta+\sin ^{7} \theta$
(d) $(\sin \theta)\left[7 \cos ^{6} \theta-35 \cos ^{4} \theta \sin ^{2} \theta+21 \cos ^{2} \theta \sin ^{4} \theta-\sin ^{6} \theta\right]$

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Key Concepts

Trigonometric Polynomials

These are expressions where sine and cosine functions are represented as polynomials in sin ? or cos ?. They arise naturally when expressing multiple-angle formulas and are useful in both simplifying trigonometric expressions and solving equations involving these functions.

Binomial Expansion

The binomial expansion is a method used to expand expressions raised to a power, such as (cos ? + i sin ?)^n. In the context of trigonometry, it allows for the systematic development of multiple-angle expressions by applying the binomial theorem to each term in the expansion.

De Moivre's Theorem

This theorem connects complex numbers and trigonometry by stating that (cos ? + i sin ?)^n equals cos(n?) + i sin(n?) for any integer n. It is a powerful tool for deriving formulas for sine and cosine of multiple angles through expansion and equating the real and imaginary parts.

Multiple-Angle Identities

These identities express trigonometric functions of multiple angles, such as sin(n?) or cos(n?), in terms of powers of the sine or cosine of a single angle. They provide a way to break down more complex trigonometric expressions into simpler components, which is particularly useful in solving equations and proving other identities.

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