Question

How do you solve the equation ${{\left( \sin x-\cos x \right)}^{2}}+\sin 2x$

Answer 1

Now to solve the given equation we will first open the bracket by the formula ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ Now we know that the value of ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ . Now we will substitute the value obtained in the given equation. Now again we will use the double angle formula for sin which is $\sin 2x=2\sin x\cos x$ . Now we will substitute the value of sin2x and simplify the equation hence we will get the value of the required equation.

Complete step by step solution:
Now consider the given expression ${{\left( \sin x-\cos x \right)}^{2}}+\sin 2x$ .
To solve the given equation we will use the algebraic properties and trigonometric properties of sin and cos.
First we will open the square bracket and simplify. We know that ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ .
Here we have $a=\sin x$ and $b=\cos x$ ,
Hence using this we get the expression as ${{\sin }^{2}}x-2\sin x\cos x+{{\cos }^{2}}x+\sin 2x$
Now rearranging the terms of the expression we get,
$\Rightarrow \left( {{\sin }^{2}}x+{{\cos }^{2}}x \right)-2\sin x\cos x+\sin 2x$
Now we know the identity ${{\sin }^{2}}x+{{\cos }^{2}}x=1$ hence using this we get,
$\Rightarrow 1-2\sin x\cos x+\sin 2x$ .
Now let us expand sin2x using the double angle formula. We know that $\sin 2x=2\sin x\cos x$ .
Hence using this formula we get the expression as
$\Rightarrow 1-\sin 2x\cos 2x+\sin 2x\cos 2x$
Hence on subtraction we get the value of the expression as 1.
Hence the value of the given expression is equal to 1.

Note:
Now note that for sin function we have $\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$ Hence on substituting the value of B as A we get the double angle formula for sin. Now we also know that $\sin \theta =\dfrac{\text{opposite side}}{\text{hypotenuse}}$ and $\cos \theta =\dfrac{\text{adjacent side}}{\text{hypotenuse}}$ . Now considering ${{\sin }^{2}}\theta +{{\cos }^{2}}\theta$ and using the Pythagoras theorem we will get the value as 1. Hence we can easily derive the two formulas used to solve the given problem.