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Question: How do you prove \[{\cos ^2}x - {\sin ^2}x = 2{\cos ^2}x - 1\]?...

How do you prove cos⁡2x−sin⁡2x=2cos⁡2x−1{\cos ^2}x - {\sin ^2}x = 2{\cos ^2}x - 1?

Explanation

Solution

Here the question is related to the trigonometry, we use the trigonometry ratios and identities we are to solve this question. In this question we have to simplify the given trigonometric ratios to its simplest form. By using the trigonometry ratios and trigonometry formulas we simplify the given trigonometric function.

Complete step by step solution:
The question is related to trigonometry and it includes the trigonometry ratios. The trigonometry ratios are sine, cosine, tangent, cosecant, secant and cotangent. In trigonometry the cosecant trigonometry ratio is the reciprocal to the sine trigonometry ratio. The secant trigonometry ratio is the reciprocal to the cosine trigonometry ratio and the cotangent trigonometry ratio is the reciprocal to the tangent trigonometry ratio.
The tangent trigonometry ratio is defined as tan⁡x=sin⁡xcos⁡x\tan x = \dfrac{{\sin x}}{{\cos x}} , The cosecant trigonometry ratio is defined as csc⁡x=1sin⁡x\csc x = \dfrac{1}{{\sin x}}, The secant trigonometry ratio is defined as sec⁡x=1cos⁡x\sec x = \dfrac{1}{{\cos x}} and The tangent trigonometry ratio is defined as cot⁡x=cos⁡xsin⁡x\cot x = \dfrac{{\cos x}}{{\sin x}}
Here we have to prove LHS = RHS
In trigonometry we have 3 trigonometry identities

⇒sin⁡2x+cos⁡2x=1 ⇒1+tan⁡2x=sec⁡2x ⇒1+cot⁡2x=csc⁡2x \Rightarrow {\sin ^2}x + {\cos ^2}x = 1 \\\ \Rightarrow 1 + {\tan ^2}x = {\sec ^2}x \\\ \Rightarrow 1 + {\cot ^2}x = {\csc ^2}x \\\

Consider LHS
cos⁡2x−sin⁡2x{\cos ^2}x - {\sin ^2}x
By the trigonometry identities, the sine trigonometry ratio is written as sin⁡2x=(1−cos⁡2x){\sin ^2}x = (1 - {\cos ^2}x)and the above equation is written as
⇒cos⁡2x−(1−cos⁡2x)\Rightarrow {\cos ^2}x - (1 - {\cos ^2}x)
On multiplying we get
⇒cos⁡2x−1+cos⁡2x\Rightarrow {\cos ^2}x - 1 + {\cos ^2}x
on simplifying we get
⇒2cos⁡2x−1\Rightarrow 2{\cos ^2}x - 1
LHS = RHS
Hence proved.
Note: Trigonometric functions are those functions that tell us the relation between the three sides of a right-angled triangle. Remember A graph is divided into four quadrants, all the trigonometric functions are positive in the first quadrant, all the trigonometric functions are negative in the second quadrant except sine and cosine functions, tangent and cotangent are positive in the third quadrant while all others are negative and similarly all the trigonometric functions are negative in the fourth quadrant except cosine and secant.