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Question: Find the value of \({(\cos x - \sin x)^2}\) , if the product \(\cos x.\sin x = 0.22\)....

Find the value of (cos⁡x−sin⁡x)2{(\cos x - \sin x)^2} , if the product cos⁡x.sin⁡x=0.22\cos x.\sin x = 0.22.

Explanation

Solution

In order to this question, to find the value of given trigonometric expression, we will first rewrite the given facts related to this question and then we will first simplify the given expression in its simplest form by applying algebraic expression to solve the question as much easily.

Complete step by step answer:
Given that, cos⁡x.sin⁡x=0.22\cos x.\sin x = 0.22.Therefore, the value of (cos⁡x−sin⁡x)2{(\cos x - \sin x)^2} : We will first change the given trigonometric expression in its simplest form by going through the algebraic formula, (a−b)2=a2+b2−2ab{(a - b)^2} = {a^2} + {b^2} - 2ab
(cos⁡x−sin⁡x)2=cos⁡2x+sin⁡2x−2cos⁡x.sin⁡x{(\cos x - \sin x)^2} = {\cos ^2}x + {\sin ^2}x - 2\cos x.\sin x
Now, as we know- (∵sin⁡2x+cos⁡2x=1)(\because {\sin ^2}x + {\cos ^2}x = 1)
(cos⁡x−sin⁡x)2=1−2cos⁡x.sin⁡x{(\cos x - \sin x)^2} = 1 - 2\cos x.\sin x
⇒(cos⁡x−sin⁡x)2=1−2×0.22 ∴(cos⁡x−sin⁡x)2=0.56\Rightarrow {(\cos x - \sin x)^2}= 1 - 2 \times 0.22 \\\ \therefore {(\cos x - \sin x)^2}= 0.56 (as cos⁡x.sin⁡x=0.22\cos x.\sin x = 0.22 is given).

Hence, the value of (cos⁡x−sin⁡x)2{(\cos x - \sin x)^2} , if the product cos⁡x.sin⁡x=0.22\cos x.\sin x = 0.22 is 0.560.56.

Note: Sometimes students make a simple mistakes like- by taking minus sign as common from the given expression (cos⁡x−sin⁡x)2{(\cos x - \sin x)^2} to make the expression as −(sin⁡x+cos⁡x)2 - {(\sin x + \cos x)^2} . But this move is wrong because we can't separate anything common until the expression has some power or exponent.