Rewrite the following expression involving trigonometry and inverse trigonometry as an algebraic expression in x. cos(2 sin- x)
Trigonometric Identities
Trigonometry in mathematics deals with the right-angled triangle’s angles and sides. By trigonometric identities, we mean the identities we use whenever we need to express the various trigonometric functions in terms of an equation.
Inverse Trigonometric Functions
Inverse trigonometric functions are the inverse of normal trigonometric functions. Alternatively denoted as cyclometric or arcus functions, these inverse trigonometric functions exist to counter the basic trigonometric functions, such as sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (cosec). When trigonometric ratios are calculated, the angular values can be calculated with the help of the inverse trigonometric functions.
![**Problem Statement:**
Rewrite the following expression involving trigonometry and inverse trigonometry as an algebraic expression in \( x \).
\[ \cos(2 \sin^{-1} x) \]
**Instructions:**
To solve this, use trigonometric identities to express the inverse trigonometric function in terms of \( x \) and rewrite the given expression in algebraic form. The key concept is using the identity for cosine of double angles and the relationship between trigonometric functions and their inverses.
**Trigonometric Identities to Use:**
1. Double Angle Formula:
\[
\cos(2\theta) = 1 - 2\sin^2\theta
\]
2. Relationship for \(\sin^{-1}(x)\):
\[
\sin(\theta) = x \implies \theta = \sin^{-1}(x)
\]
3. From the Pythagorean identity:
\[
\cos^2(\theta) = 1 - \sin^2(\theta)
\]
Substitute \(\sin(\theta) = x\) to find \(\cos(\theta)\).
By systematically applying these identities, you will be able to derive an algebraic expression in terms of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba3a7503-ff4c-497e-ae71-1de9bb8b8d56%2F895e37dc-1222-4729-9c15-a0b7d1394f96%2F5rhayph_processed.jpeg&w=3840&q=75)
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