{"id":1973,"date":"2025-07-31T21:41:47","date_gmt":"2025-07-31T21:41:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=1973"},"modified":"2026-03-24T17:26:48","modified_gmt":"2026-03-24T17:26:48","slug":"simplifying-trigonometric-expressions-with-identities-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/simplifying-trigonometric-expressions-with-identities-learn-it-3\/","title":{"raw":"Simplifying Trigonometric Expressions With Identities: Learn It 3","rendered":"Simplifying Trigonometric Expressions With Identities: Learn It 3"},"content":{"raw":"

Simplify trigonometric expressions using algebra and the identities<\/h2>\r\nWe have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.\r\n\r\nFor example, the equation [latex]\\left(\\sin x+1\\right)\\left(\\sin x - 1\\right)=0[\/latex] resembles the equation [latex]\\left(x+1\\right)\\left(x - 1\\right)=0[\/latex], which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.\r\n\r\nAnother example is the difference of squares formula, [latex]{a}^{2}-{b}^{2}=\\left(a-b\\right)\\left(a+b\\right)[\/latex], which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.\r\n\r\n
Write the following trigonometric expression as an algebraic expression: [latex]2{\\cos }^{2}\\theta +\\cos \\theta -1[\/latex].[reveal-answer q=\"19423\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"19423\"]Notice that the pattern displayed has the same form as a standard quadratic expression, [latex]a{x}^{2}+bx+c[\/latex]. Letting [latex]\\cos \\theta =x[\/latex], we can rewrite the expression as follows:\r\n

[latex]2{x}^{2}+x - 1[\/latex]<\/p>\r\nThis expression can be factored as [latex]\\left(2x+1\\right)\\left(x - 1\\right)[\/latex]. If it were set equal to zero and we wanted to solve the equation, we would use the zero factor property and solve each factor for [latex]x[\/latex]. At this point, we would replace [latex]x[\/latex] with [latex]\\cos \\theta [\/latex] and solve for [latex]\\theta [\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

Rewrite the trigonometric expression: [latex]4{\\cos }^{2}\\theta -1[\/latex].[reveal-answer q=\"947607\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"947607\"]Notice that both the coefficient and the trigonometric expression in the first term are squared, and the square of the number 1 is 1. This is the difference of squares. Thus,\r\n

[latex]\\begin{align}4{\\cos }^{2}\\theta -1&={\\left(2\\cos \\theta \\right)}^{2}-1 \\\\ &=\\left(2\\cos \\theta -1\\right)\\left(2\\cos \\theta +1\\right) \\end{align}[\/latex]<\/p>\r\n\r\n

Analysis<\/h3>\r\nIf this expression were written in the form of an equation set equal to zero, we could solve each factor using the zero factor property. We could also use substitution like we did in the previous problem and let [latex]\\cos \\theta =x[\/latex], rewrite the expression as [latex]4{x}^{2}-1[\/latex], and factor [latex]\\left(2x - 1\\right)\\left(2x+1\\right)[\/latex]. Then replace [latex]x[\/latex] with [latex]\\cos \\theta [\/latex] and solve for the angle.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>\r\n
\r\n
\r\n\r\nRewrite the trigonometric expression: [latex]25 - 9{\\sin }^{2}\\theta [\/latex].\r\n\r\n[reveal-answer q=\"979877\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"979877\"]\r\n\r\nThis is a difference of squares formula: [latex]25 - 9{\\sin }^{2}\\theta =\\left(5 - 3\\sin \\theta \\right)\\left(5+3\\sin \\theta \\right)[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n
Simplify the expression by rewriting and using identities:\r\n

[latex]{\\csc }^{2}\\theta -{\\cot }^{2}\\theta [\/latex]<\/p>\r\n[reveal-answer q=\"120852\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"120852\"]\r\n\r\nWe can start with the Pythagorean identity.\r\n

[latex]1+{\\cot }^{2}\\theta ={\\csc }^{2}\\theta [\/latex]<\/p>\r\nNow we can simplify by substituting [latex]1+{\\cot }^{2}\\theta [\/latex] for [latex]{\\csc }^{2}\\theta [\/latex]. We have\r\n

[latex]\\begin{align}{\\csc }^{2}\\theta -{\\cot }^{2}\\theta &=1+{\\cot }^{2}\\theta -{\\cot }^{2}\\theta \\\\ &=1\\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

[ohm_question hide_question_numbers=1]120496[\/ohm_question]<\/section>
Use algebraic techniques to verify the identity: [latex]\\frac{\\cos\\theta}{1+\\sin\\theta}=\\frac{1-\\sin\\theta}{\\cos\\theta}[\/latex].(Hint: Multiply the numerator and denominator on the left side by [latex]1-\\sin\\theta[\/latex]).[reveal-answer q=\"94618\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"94618\"]\r\n

[latex]\\begin{align}\\frac{\\cos \\theta }{1+\\sin \\theta }\\left(\\frac{1-\\sin \\theta }{1-\\sin \\theta }\\right)&=\\frac{\\cos \\theta \\left(1-\\sin \\theta \\right)}{1-{\\sin }^{2}\\theta } \\\\ &=\\frac{\\cos \\theta \\left(1-\\sin \\theta \\right)}{{\\cos }^{2}\\theta } \\\\ &=\\frac{1-\\sin \\theta }{\\cos \\theta } \\end{align}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

Simplify trigonometric expressions using algebra and the identities<\/h2>\n

We have seen that algebra is very important in verifying trigonometric identities, but it is just as critical in simplifying trigonometric expressions before solving. Being familiar with the basic properties and formulas of algebra, such as the difference of squares formula, the perfect square formula, or substitution, will simplify the work involved with trigonometric expressions and equations.<\/p>\n

For example, the equation [latex]\\left(\\sin x+1\\right)\\left(\\sin x - 1\\right)=0[\/latex] resembles the equation [latex]\\left(x+1\\right)\\left(x - 1\\right)=0[\/latex], which uses the factored form of the difference of squares. Using algebra makes finding a solution straightforward and familiar. We can set each factor equal to zero and solve. This is one example of recognizing algebraic patterns in trigonometric expressions or equations.<\/p>\n

Another example is the difference of squares formula, [latex]{a}^{2}-{b}^{2}=\\left(a-b\\right)\\left(a+b\\right)[\/latex], which is widely used in many areas other than mathematics, such as engineering, architecture, and physics. We can also create our own identities by continually expanding an expression and making the appropriate substitutions. Using algebraic properties and formulas makes many trigonometric equations easier to understand and solve.<\/p>\n

Write the following trigonometric expression as an algebraic expression: [latex]2{\\cos }^{2}\\theta +\\cos \\theta -1[\/latex].<\/p>\n