Question

How do you multiply and simplify \[{{\left( \sin x-\cos x \right)}^{2}}\]?

Answer 1

Hint: This question is from the trigonometry. In this question, we will simplify the term \[{{\left( \sin x-\cos x \right)}^{2}}\]. In solving this question, we will first separate the terms which are given in the question. After that, we will multiply them using the formula \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\]. After that, we will see some formulas of trigonometry and use them after multiplying them. After solving the further question, we will answer. After that, we will see another method of multiplying.

Complete step-by-step solution:
Let us solve this question.
In this question, we have asked to multiply and simplify the given term. The given term is:
\[{{\left( \sin x-\cos x \right)}^{2}}\]
As we know that \[{{x}^{2}}\] can also be written as \[x\cdot x\], so we can write the above term as
\[{{\left( \sin x-\cos x \right)}^{2}}=\left( \sin x-\cos x \right)\left( \sin x-\cos x \right)\]
Now, using the formula of foil method that is: \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\], we can write
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}=\left( \sin x\cdot \sin x-\sin x\cdot \cos x-\sin x\cdot \cos x+\cos x\cdot \cos x \right)\]
The above equation can also be written as
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}=\left( {{\sin }^{2}}x-2\cdot \sin x\cdot \cos x+{{\cos }^{2}}x \right)\]
The above equation can also be written as
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}=\left( {{\sin }^{2}}x+{{\cos }^{2}}x-2\cdot \sin x\cdot \cos x \right)\]
Now, using the formula of trigonometry: \[{{\sin }^{2}}x+{{\cos }^{2}}x=1\], we can write
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}=\left( 1-2\cdot \sin x\cdot \cos x \right)\]
Now, using the formula of trigonometry: \[2\cdot \sin x\cdot \cos x=\sin \left( 2x \right)\], we can write
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}=\left( 1-\sin \left( 2x \right) \right)\]
Hence, we have multiplied and simplified the term \[{{\left( \sin x-\cos x \right)}^{2}}\]. The simplified value of \[{{\left( \sin x-\cos x \right)}^{2}}\] is \[\left( 1-\sin \left( 2x \right) \right)\].

Note: We should have a better knowledge in the topic of trigonometry. We should remember the following formulas to solve this type of question easily:
\[2\cdot \sin x\cdot \cos x=\sin \left( 2x \right)\]
\[{{\sin }^{2}}x+{{\cos }^{2}}x=1\]
Remember the foil method that is: \[\left( a+b \right)\left( c+d \right)=ac+ad+bc+bd\]
We can multiply the term by a different method.
For that method, we will have to know a formula. The formula is:
\[{{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab\]
Now, solve this question.
So, using the above formula, we can write the term \[{{\left( \sin x-\cos x \right)}^{2}}\] as
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}={{\sin }^{2}}x+{{\cos }^{2}}x-2\cdot \sin x\cdot \cos x\]
Now, we will use the formulas of trigonometry here that we have seen above. We can write
\[\Rightarrow {{\left( \sin x-\cos x \right)}^{2}}=1-\sin \left( 2x \right)\]
Hence, we can solve this question by this method too.