Question

The value of \[\left[ {{{\cos }^4}A - {{\sin }^4}A} \right]\] is equal to:
A) \[2{\cos ^2}A + 1\]
B) \[2{\cos ^2}A - 1\]
C) \[2{\sin ^2}A - 1\]
D) \[2{\sin ^2}A + 1\]

Answer 1

Hint:
In the given question, we have been asked to find the value of a polynomial consisting of two different variables. The polynomial is clearly of degree four. The first term is a variable raised to fourth power and the second variable is also raised to the same power. So, we have to apply the formula of difference of two squares, twice. Then we are going to use the appropriate formulae and solve for it.

Formula Used:
We are to apply the formula of difference of two squares:
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]

Complete step by step solution:
We have to find the value of \[p = {\cos ^4}A - {\sin ^4}A\].
Clearly, \[{\cos ^4}A - {\sin ^4}A = {\left( {{{\cos }^2}A} \right)^2} - {\left( {{{\sin }^2}A} \right)^2}\]
We are to apply the formula of difference of two squares:
\[{a^2} - {b^2} = \left( {a + b} \right)\left( {a - b} \right)\]
\[p = \left( {{{\cos }^2}A - {{\sin }^2}A} \right)\left( {{{\cos }^2}A + {{\sin }^2}A} \right)\]
We know, \[{\cos ^2}A + {\sin ^2}A = 1\]
Hence, \[p = {\cos ^2}A - {\sin ^2}A\]
Again, \[{\sin ^2}A = 1 - {\cos ^2}A\]
So, \[p = {\cos ^2}A - \left( {1 - {{\cos }^2}A} \right) = {\cos ^2}A - 1 + {\cos ^2}A\]
Hence, \[p = 2{\cos ^2}A - 1\]

Thus, the correct option is B.

Additional Information:
We only have the simplified formula of difference of two squares, \[{a^2} - {b^2}\]. We do not have the formula for the sum of two squares, \[{a^2} + {b^2}\]. For that we have a combination of two formulae – \[{a^2} + {b^2} = \left( {{{\left( {a + b} \right)}^2} + {{\left( {a - b} \right)}^2}} \right)/2\].

Note:
When the numbers are raised to some power whose formula is not known, we try to break it down to the ones whose formula is known and solve it accordingly. Here, we do not know the formula of difference of fourth power of two numbers, so we first reduced it to the ones whose formula we know, then we evaluated it, used the standard trigonometric identities and simplified the answer using their result.