Enter an angle and select its unit (degrees or radians) to apply the power reducing identities. Your results show the reduced forms of sin²(x), cos²(x), and tan²(x) — each expressed in terms of cos(2x) — along with the values of sin(x), cos(x), and tan(x) at your angle.
sin²(x)
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cos²(x)
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tan²(x)
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sin(x)
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cos(x)
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tan(x)
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cos(2x)
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Power reducing refers to the use of trigonometric identities that rewrite squared (or higher-power) trig functions in terms of first-power trig functions of a doubled angle. For example, sin²(x) is rewritten as (1 − cos(2x)) / 2. This simplifies integration, differentiation, and algebraic manipulation of trig expressions.
The three fundamental power reducing formulas are: sin²(x) = (1 − cos(2x)) / 2, cos²(x) = (1 + cos(2x)) / 2, and tan²(x) = (1 − cos(2x)) / (1 + cos(2x)). These are derived from the double-angle identity for cosine: cos(2x) = 1 − 2sin²(x) = 2cos²(x) − 1.
To reduce the power, identify the squared trig function in your expression, then substitute the appropriate power-reducing formula. For instance, to simplify sin²(x), replace it with (1 − cos(2x)) / 2. If you have higher powers like sin⁴(x), write it as (sin²(x))² and apply the formula twice, expanding as needed.
cos⁴(x) = (cos²(x))² = ((1 + cos(2x)) / 2)² = (1 + 2cos(2x) + cos²(2x)) / 4. Applying the power reducing formula again to cos²(2x) = (1 + cos(4x)) / 2 gives: cos⁴(x) = (3 + 4cos(2x) + cos(4x)) / 8.
Power reducing identities are widely used in calculus — especially when integrating even powers of sine or cosine — as well as in signal processing, electrical engineering, and physics. Real-time systems such as space research and navigation software rely on these identities to compute trigonometric functions efficiently at higher powers.
The three Pythagorean identities are: sin²(x) + cos²(x) = 1, 1 + tan²(x) = sec²(x), and 1 + cot²(x) = csc²(x). These form the foundation of many trigonometric simplifications and are closely related to the power reducing formulas.
The six trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Power reducing identities are most commonly applied to sin, cos, and tan, since those appear most frequently in calculus and applied mathematics.
Yes, the power reducing formulas work identically for angles measured in radians or degrees. Simply ensure your calculator or computation uses the correct unit throughout. This calculator supports both — just select your preferred unit from the dropdown before entering your angle.