Prove the following for all x ∈ℝ : (i) sin2 x=2 sinx cosx, (ii) cos2 x=cos^2 x-sin^2 x=2 cos^2 x

step 1 here the LHS is given as sine 3 pi by 2 plus x plus sine 3 pi by 2 minus x is will be equal to f

Prove the following for all x ∈ℝ : (i) sin2 x=2 sinx cosx,...

Prove the following for all $x \in \mathbb{R}$ : (i) $\sin 2 x=2 \sin x \cos x$, (ii) $\cos 2 x=\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1=1-2 \sin ^{2} x$, (iii) $\sin 3 x=3 \sin x-4 \sin ^{3} x$ (iv) $\cos 3 x=4 \cos ^{3} x-3 \cos x$. Deduce that $\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}=\cos \frac{\pi}{4}, \quad \sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}=\cos \frac{\pi}{6}, \quad \cos \frac{\pi}{3}=\frac{1}{2}=\sin \frac{\pi}{6}$

Prove the following for all $x \in \mathbb{R}$ :
(i) $\sin 2 x=2 \sin x \cos x$,
(ii) $\cos 2 x=\cos ^{2} x-\sin ^{2} x=2 \cos ^{2} x-1=1-2 \sin ^{2} x$,
(iii) $\sin 3 x=3 \sin x-4 \sin ^{3} x$
(iv) $\cos 3 x=4 \cos ^{3} x-3 \cos x$.
Deduce that
$\sin \frac{\pi}{4}=\frac{1}{\sqrt{2}}=\cos \frac{\pi}{4}, \quad \sin \frac{\pi}{3}=\frac{\sqrt{3}}{2}=\cos \frac{\pi}{6}, \quad \cos \frac{\pi}{3}=\frac{1}{2}=\sin \frac{\pi}{6}$

Key Concepts

Addition Formulas for Trigonometric Functions

The addition formulas, such as sin(a + b) = sin a cos b + cos a sin b and cos(a + b) = cos a cos b ? sin a sin b, are fundamental identities in trigonometry that allow for the derivation of other important identities. They enable the expression of trigonometric functions of sums of angles in terms of functions of individual angles, forming the basis for proving the double and triple angle identities.

Double Angle Identities

Double angle identities are specific cases of the addition formulas where the two angles are equal, such as in sin 2x = 2 sin x cos x and cos 2x = cos^2 x ? sin^2 x. These identities simplify expressions by representing functions of twice an angle in terms of functions of the single angle, and they also have alternate forms depending on expressing one function in terms of the other.

Triple Angle Identities

Triple angle identities, like sin 3x = 3 sin x ? 4 sin^3 x and cos 3x = 4 cos^3 x ? 3 cos x, extend the idea of addition formulas to the case where an angle is added three times. These identities provide more complex relationships that involve powers of sine and cosine, and they are often derived by using the double angle and addition formulas in a sequential manner.

Special Angle Values in Trigonometry

Special angle values, such as those for ?/4, ?/3, and ?/6, are determined through geometric considerations, often involving special right triangles or the unit circle. These specific values are used to compute exact numerical values of sine and cosine, and they serve as benchmarks in both theoretical problems and practical applications in trigonometry.

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