(a) (i) Use de Moivre's theorem to express cos 40 and sin 40 in terms of cos 0 and sin 0. 4 tan3 0 4 tan 0 – (ii) Hence prove that tan 40 1 - 6 tan² 0 + tan4 0` (iii) Deduce that tan' -1 24 -1 1 4 tan 7 3'

Q: sin 2x Verify 1+ cos 2x = tan x.

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Q: 1- tan- (0) = 2 cos? (0) – 1 1+ tan“ (0)

A:

Q: 3 Prove the identity: sin30 = sin® –sin30 using De Moivre's theorem.

A: To prove the given identity ,we have to use De Moivre's identity.

Q: (a) i) Prove that 1 sin20 1 tane 1+ sin20 1+ tane for 0<0 <- 4 ii) Hence, show that the exact value…

A:

Q: (2) Find the exact value of the following: (i) cos-(-) (ii) tan-V3 (ii) 2tan-1+ tan-1, (iv) sec…

A: (i) cos-1(-1/2) = 2π/3 (ii)tan-1(√3) = π/3

Q: Show that the equation 2x + cos(x) = 1 has exactly one root. (IVT and Rolle's Theorem)

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Q: Prove that

A: Given:d2ndx2nsinx=-1n sinxd2ndx2ncosx=-1n cosx

Q: cos A+1 cSc A %3D sin' A 3 1-cos A

A: Consider the given identity, cosA+1sin3A=cscA1-cosA Cross multiply the equation,…

Q: sin' t – cos' t Prove tan?t – 1 cos? t

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Q: Part D

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Q: Q2/ A) Prove the identity Sin (A + B) + Sin (A - B ) = 2SinA cosB

A: Given: sinA+B+sinA-B=2 sin A cos B To Find: To prove the given inequality.

Q: 4) Verify. cos²(x)[1 C. – tan²(x)] = cos(2x)

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Q: Prove identity (1+ a) (1 – cos a) = tan? cos æ) sec x x cos x

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Q: Prove tan x + cos x = sin x (sec x + cot x).

A: We have to prove the identity: tan x+cos x=sin x sec x+cot x We know the relation between…

Q: cos(x) 3) 1-sin(x) sec(x) + tan(x)

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Q: 11) Prove each identity: a) cos(a+b) cos(a)cos (b) b) 2 sec2(t)= - 1 tan(a) tan (b) 1-sin(t) cos²(t)…

A: a.cos(a)cos(b)cos(a+b)​=1−tan(a)tan(b)Note: For tan(a)tan(b), we have the trigonometric identity…

Q: sin 0(3sin? 0+ 4cos? 0) = 3 cos 0 + cos³ 0 4. Prove that tan 0

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Q: 2) Find the EXACT value of cos if tan(0) = 3 and sin(0) < 0

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Vertices Of An Ellipse

In the study of the conic section of the geometry, An ellipse is a plane curve surrounding two focal points, or foci where the set of all locus of points in the plane has their sum of the distances to the two foci is constant. On the ellipse curve, it is the intersection point of the main axis. In an ellipse graph, the center of the axis is the midpoint of the line segment connecting the foci. The major axis is the line section that runs through the ellipse's foci, while the minor axis runs through the middle and is perpendicular to the major axis.. The ellipse has two endpoints on the major axis which are known as the vertices (in plural), vertex in the singular. The equation of the major axis is 2a, the equation of the minor is 2b and the distance between the major axis and minor axis is 2c. The semi-major axis has a length of a and the semi-minor axis has a length of b. The ellipse's vertices are determined by the elliptical orbit's equation and graph.