Prove that:\(\cos^{-1}(4\mathrm x^3-3\mathrm x)=3\cos^{-1}\mathrm x,\f

1.	Prove that:\(\cos^{-1}(4\mathrm x^3-3\mathrm x)=3\cos^{-1}\mathrm x,\frac{1}{2}\leq\mathrm x\leq1\)
Answer» To Prove: cos ^-1(4x ³– 3x) = 3 cos ^-1 x Formula Used: cos 3A = 4 cos ³ A – 3 cos A Proof: LHS = cos^-1 (4x ³– 3x) … (1) Let x = cos A … (2) Substituting (2) in (1), LHS = cos ^-1 (4 cos ³A – 3 cos A) = cos ^-1 (cos 3A) = 3A From (2), A = cos ^-1 x, 3A = 3 cos ^-1 x = RHS Therefore, LHS = RHS Hence proved.

Prove that:\(\cos^{-1}(4\mathrm x^3-3\mathrm x)=3\cos^{-1}\mathrm x,\frac{1}{2}\leq\mathrm x\leq1\)

Answer»

To Prove:

cos ^-1(4x ³– 3x) = 3 cos ^-1 x

Formula Used: cos 3A = 4 cos ³ A – 3 cos A

Proof:

LHS = cos^-1 (4x ³– 3x) … (1)

Let x = cos A … (2)

Substituting (2) in (1),

LHS = cos ^-1 (4 cos ³A – 3 cos A)

= cos ^-1 (cos 3A)

= 3A

From (2), A = cos ^-1 x,

3A = 3 cos ^-1 x

= RHS

Therefore, LHS = RHS

Hence proved.