If the following exression represents an identity 1-sin2x÷cos2x =A then value of A is given by
The Correct Answer and Explanation is:
We are given the trigonometric expression:1−sin2xcos2x=A\frac{1 – \sin 2x}{\cos 2x} = Acos2x1−sin2x=A
We are to find the value of AAA and explain the identity.
Step-by-step Solution:
We start by recalling a useful trigonometric identity:tanθ=sinθcosθ\tan \theta = \frac{\sin \theta}{\cos \theta}tanθ=cosθsinθ
But that does not directly help. Instead, consider separating the terms in the expression:A=1−sin2xcos2xA = \frac{1 – \sin 2x}{\cos 2x}A=cos2x1−sin2x
There is no direct simplification unless we recall that sine and cosine of double angles can be expressed using basic identities, but since there is no simplification of 1−sin2×1 – \sin 2×1−sin2x in a way that would cancel with cos2x\cos 2xcos2x, we try another approach.
We can try breaking this into separate terms using basic algebra:A=1cos2x−sin2xcos2xA = \frac{1}{\cos 2x} – \frac{\sin 2x}{\cos 2x}A=cos2x1−cos2xsin2x
This separates the original expression into two parts:A=sec2x−tan2xA = \sec 2x – \tan 2xA=sec2x−tan2x
So, the simplified form of the expression is:sec2x−tan2x\boxed{\sec 2x – \tan 2x}sec2x−tan2x
Final Answer:
A=sec2x−tan2xA = \boxed{\sec 2x – \tan 2x}A=sec2x−tan2x
Explanation:
The given expression is:1−sin2xcos2x\frac{1 – \sin 2x}{\cos 2x}cos2x1−sin2x
To simplify this expression and find what it is equal to, we use algebraic manipulation and trigonometric identities. The first step is to break the single fraction into two separate terms. We write:1−sin2xcos2x=1cos2x−sin2xcos2x\frac{1 – \sin 2x}{\cos 2x} = \frac{1}{\cos 2x} – \frac{\sin 2x}{\cos 2x}cos2x1−sin2x=cos2x1−cos2xsin2x
Now, look at each term separately. The first term, 1cos2x\frac{1}{\cos 2x}cos2x1, is a standard trigonometric identity. It is equal to sec2x\sec 2xsec2x, which is the reciprocal of cosine. The second term, sin2xcos2x\frac{\sin 2x}{\cos 2x}cos2xsin2x, is also a known identity. It simplifies to tan2x\tan 2xtan2x, since tangent is the ratio of sine to cosine.
Combining both simplified expressions gives us:sec2x−tan2x\sec 2x – \tan 2xsec2x−tan2x
This means the original expression is equivalent to this simplified trigonometric form. Therefore, the identity is:1−sin2xcos2x=sec2x−tan2x\frac{1 – \sin 2x}{\cos 2x} = \sec 2x – \tan 2xcos2x1−sin2x=sec2x−tan2x
Identities like these are important in trigonometry because they allow us to rewrite complex expressions in simpler or more familiar forms. These simplifications help solve equations, integrate functions, or analyze wave behavior. Understanding these identities also strengthens your skill in recognizing patterns and relationships between trigonometric functions.
