   Question Answers

# Prove that$\sin 3A.{\sin ^3}A + \cos 3A.{\cos ^3}A = {\cos ^3}2A$  Hint: We cannot solve the Question directly , we have to use some identities, which identity is to be used. We can get a hint of it from the question itself like cos3A whose identity is known to us, simplify the equation and see if you can get something common out and proceed.

We know that
$\cos 3A = 4{\cos ^3}A - 3\cos A \\ \sin 3A = 3\sin A - 4{\sin ^3}A \\$

Using these in L.H.S
$\therefore$ The L.H.S
= ${\sin ^3}A\left( {3\sin A - 4{{\sin }^3}A} \right) + {\cos ^3}A\left( {4{{\cos }^3}A - 3\cos A} \right)$
= $3{\sin ^4}A - 4{\sin ^6}A + 4{\cos ^6}A - 3{\cos ^4}A$

Taking common
$= - 3\left( {{{\cos }^4}A - {{\sin }^4}A} \right) + 4\left( {{{\cos }^6}A - {{\sin }^6}A} \right) \\ = 4\left( {{{\left( {{{\cos }^2}A} \right)}^3} - {{\left( {{{\sin }^2}A} \right)}^3}} \right) - 3\left( {{{\left( {{{\cos }^2}A} \right)}^2} - {{\left( {{{\sin }^2}A} \right)}^2}} \right) \\$

Using basic algebraic formula,
$= 4\left( {{{\cos }^2}A - {{\sin }^2}A} \right)\left( {{{({{\cos }^2}A)}^2} + {{\left( {{{\sin }^2}A} \right)}^2} + {{\cos }^2}A{{\sin }^2}A} \right) - 3\left( {{{\cos }^2}A - {{\sin }^2}A} \right)\left( {{{\cos }^2}A + {{\sin }^2}A} \right)$

We know that ${\sin ^2}A + {\cos ^2}A = 1$,${\cos ^2}A - {\sin ^2}A = \cos 2A$
And taking common $\left( {{{\cos }^2}A - {{\sin }^2}A} \right)$
$= \cos 2A\left( {4{{\cos }^4}A + 4{{\sin }^4}A + 4{{\cos }^2}A{{\sin }^2}A - 3.1} \right)$

We can replace 1 by ${\left( {{{\sin }^2}A + {{\cos }^2}A} \right)^2}$
$= \cos 2A\left( {4{{\cos }^4}A + 4{{\sin }^4}A + 4{{\cos }^2}A{{\sin }^2}A - 3.{{\left( {{{\sin }^2}A + {{\cos }^2}A} \right)}^2}} \right)$$= \cos 2A\left( {4{{\cos }^4}A + 4{{\sin }^4}A + 4{{\cos }^2}A{{\sin }^2}A - 3.{{\left( {{{\sin }^4}A + {{\cos }^4}A + 2{{\sin }^2}A{{\cos }^2}A} \right)}^{}}} \right)$$= \cos 2A\left( {4{{\cos }^4}A + 4{{\sin }^4}A + 4{{\cos }^2}A{{\sin }^2}A - 3{{\sin }^4}A - 3{{\cos }^4}A - 6{{\cos }^2}A{{\sin }^2}A} \right)$
$= \cos 2A\left( {{{\cos }^4}A + {{\sin }^4}A - 2{{\cos }^2}A{{\sin }^2}A} \right) \\ = \cos 2A{\left( {{{\cos }^2}A - {{\sin }^2}A} \right)^2} \\ = \cos 2A.{(\cos 2A)^2} \\ = {\cos ^3}2A \\$
=R.H.S

Note: Before solving such questions one needs to have a fine knowledge of trigonometry. About the properties of sine and cosines. Observe the question carefully the given terms will be like those in the properties, then only you will get to know what to apply.
Sin Cos Formula  Cos 360  Value of Cos 120  CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry Formula  Sin 120  Value of Sin 180  CBSE Class 10 Maths Chapter 10 - Circles Formula  Trigonometry Table  CBSE Class 10 Maths Chapter 2 - Polynomials Formula  CBSE Class 10 Maths Chapter 14 - Statistics Formula  Important Questions for CBSE Class 10 Maths Chapter 8 - Introduction to Trigonometry  Important Questions for CBSE Class 10 Maths Chapter 9 - Some Applications of Trigonometry  Important Questions for CBSE Class 10 Maths Chapter 10 - Circles  Important Questions for CBSE Class 10 Maths & Science with Answers  Important Questions for CBSE Class 10 Maths Chapter 2 - Polynomials  Important Questions for CBSE Class 10 Maths Chapter 11 - Constructions  Important Questions for CBSE Class 10 Maths Chapter 6 - Triangles  Important Questions for CBSE Class 10 Maths Chapter 15 - Probability  Important Questions for CBSE Class 9 Maths Chapter 10 - Circles  Important Questions for CBSE Class 6 Maths Chapter 10 - Mensuration  CBSE Class 10 Maths Question Paper 2017  CBSE Class 10 Maths Question Paper 2020  Maths Question Paper for CBSE Class 10 - 2011  Maths Question Paper for CBSE Class 10 - 2008  Maths Question Paper for CBSE Class 10 - 2012  Maths Question Paper for CBSE Class 10 (2016)  Maths Question Paper for CBSE Class 10 - 2009  Maths Question Paper for CBSE Class 10 - 2010  Maths Question Paper for CBSE Class 10 - 2007  CBSE Class 10 Maths Question Paper 2018 - Free PDF  RS Aggarwal Solutions Class 10 Chapter 3 - Linear Equations in two variables (Ex 3A) Exercise 3.1  RS Aggarwal Solutions Class 7 Chapter-3 Decimals (Ex 3A) Exercise 3.1  RS Aggarwal Solutions Class 6 Chapter-3 Whole Numbers (Ex 3A) Exercise 3.1  RS Aggarwal Solutions Class 8 Chapter-3 Squares and Square Roots (Ex 3A) Exercise 3.1  NCERT Solutions for Class 10 Maths Chapter 8 Introduction to Trigonometry  NCERT Solutions for Class 10 Maths Chapter 9 Some Applications of Trigonometry  RD Sharma Solutions for Class 10 Maths Chapter 12 - Some Applications of Trigonometry  NCERT Solutions for Class 10 English Footprints Without Feet Chapter 10 - The Book that Saved the Earth  NCERT Solutions for Class 10 Maths Chapter 8 - Introduction to Trigonometry  RS Aggarwal Solutions Class 10 Chapter 10 - Quadratic Equations (Ex 10C) Exercise 10.3  