Simplify (1 - Sin x)(1 + Sin x)For all conceivable values of the variables inside their respective domains, trigonometric identities are equations using trigonometric functions that are true.. These identities are derived from the geometric relationships and properties of right triangles, circles, and periodic functions. In this article, we will derive one such trigonometric identity by simplifying (1 - Sin x)(1 + Sin x). What are Trigonometric RatiosTrigonometric ratios are used to quantitatively connect the angles of a right triangle to the ratios of the lengths of its sides. Six fundamental trigonometric ratios exist in trigonometry: sine, cosine, tangent, cosecant, secant, and cotangent. These ratios are frequently represented by the abbreviations sin, cos, tan, csc, sec, and cot. The sine function is denoted as "Sin" and is defined as follows: The cosine function is denoted as "Cos" and is defined as follows: Here, theta represents the angle, and the opposite and adjacent sides refer to the sides of the right triangle relative to the angle. What Is (1 - Sin x)(1 + Sin x) Mathematically, we can say that: (1 - Sin x)(1 + Sin x)= Cos^{2} x This identity provides a way to express the sine of an angle in terms of square of its cosine. It helps simplify trigonometric expressions by replacing the tangent of an angle with the cosine of the angle. ExplanationTo prove that (1 - Sin x)(1 + Sin x) is equal to Cos^{2} x, we can start with the left side of the equation and manipulate it using algebraic identities until we arrive at the right side. Here's a step-by-step explanation:
Thus, we have proved that (1 - Sin x)(1 + Sin x) is equal to cos^{2} x by stepwise manipulation of the equation. Uses of Trigonometric IdentitiesThe trigonometric identities are immensely useful in trigonometry, calculus, and physics.
Conclusion(1 - Sin x)(1 + Sin x)= Cos^{2} x is a fundamental trigonometric identity that relates the sine function to the square of cosine of the angle. It serves as a powerful tool in trigonometry, calculus, and physics, facilitating the simplification of expressions, solving equations, and analyzing geometric and wave-related problems. Its versatility and wide range of applications make it an essential concept for anyone studying or working with trigonometry and its related fields. |