Worksheet of trigonometric identities . we also use algebraic ... - Free Printable
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Step-by-step solution for: Worksheet of trigonometric identities . we also use algebraic ...
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Show Answer Key & Explanations
Step-by-step solution for: Worksheet of trigonometric identities . we also use algebraic ...
1. Start with the left side: tan(x)sin(x) + cos(x). Rewrite tan(x) as sin(x)/cos(x): (sin(x)/cos(x)) * sin(x) + cos(x) = sin²(x)/cos(x) + cos(x). Combine terms over a common denominator: (sin²(x) + cos²(x)) / cos(x). Use the Pythagorean identity sin²(x) + cos²(x) = 1: 1/cos(x) = sec(x). This equals the right side.
2. Start with the left side: 1/tan(x) + tan(x). Rewrite 1/tan(x) as cot(x): cot(x) + tan(x). Express both in terms of sin and cos: cos(x)/sin(x) + sin(x)/cos(x). Combine over a common denominator: (cos²(x) + sin²(x)) / (sin(x)cos(x)). Use the Pythagorean identity: 1 / (sin(x)cos(x)). This equals the right side.
3. Start with the left side: sin(x) - sin(x)cos²(x). Factor out sin(x): sin(x)(1 - cos²(x)). Use the Pythagorean identity 1 - cos²(x) = sin²(x): sin(x) * sin²(x) = sin³(x). This equals the right side.
4. Start with the left side: cos(α)/(1 + sin(α)) + (1 + sin(α))/cos(α). Combine over a common denominator: [cos²(α) + (1 + sin(α))²] / [cos(α)(1 + sin(α))]. Expand the numerator: cos²(α) + 1 + 2sin(α) + sin²(α). Group terms: (cos²(α) + sin²(α)) + 1 + 2sin(α) = 1 + 1 + 2sin(α) = 2 + 2sin(α) = 2(1 + sin(α)). The expression becomes: 2(1 + sin(α)) / [cos(α)(1 + sin(α))]. Cancel (1 + sin(α)): 2/cos(α) = 2sec(α). This equals the right side.
5. Start with the left side: cos(x)/(1 - sin(x)) - cos(x)/(1 + sin(x)). Combine over a common denominator: [cos(x)(1 + sin(x)) - cos(x)(1 - sin(x))] / [(1 - sin(x))(1 + sin(x))]. Factor cos(x) in the numerator: cos(x)[(1 + sin(x)) - (1 - sin(x))] = cos(x)[1 + sin(x) - 1 + sin(x)] = cos(x)[2sin(x)] = 2sin(x)cos(x). The denominator is a difference of squares: 1 - sin²(x) = cos²(x). The expression is: 2sin(x)cos(x) / cos²(x) = 2sin(x)/cos(x) = 2tan(x). This equals the right side.
6. Start with the right side: (csc(x)cos(x)) / (tan(x) + cot(x)). Rewrite all functions in terms of sin and cos: ( (1/sin(x)) * cos(x) ) / ( sin(x)/cos(x) + cos(x)/sin(x) ). Simplify numerator: cos(x)/sin(x). Denominator: (sin²(x) + cos²(x)) / (sin(x)cos(x)) = 1 / (sin(x)cos(x)). The expression becomes: (cos(x)/sin(x)) / (1/(sin(x)cos(x))) = (cos(x)/sin(x)) * (sin(x)cos(x)/1) = cos²(x). This equals the left side.
7. Start with the left side: (sin⁴(x) - cos⁴(x)) / (sin²(x) - cos²(x)). Factor the numerator as a difference of squares: (sin²(x) - cos²(x))(sin²(x) + cos²(x)). The expression becomes: [(sin²(x) - cos²(x))(sin²(x) + cos²(x))] / (sin²(x) - cos²(x)). Cancel (sin²(x) - cos²(x)): sin²(x) + cos²(x) = 1. This equals the right side.
8. Start with the left side: tan²(x) / (tan²(x) + 1). Use the Pythagorean identity tan²(x) + 1 = sec²(x): tan²(x) / sec²(x). Rewrite in terms of sin and cos: (sin²(x)/cos²(x)) / (1/cos²(x)) = (sin²(x)/cos²(x)) * (cos²(x)/1) = sin²(x). This equals the right side.
9. Start with the left side: (1 - sin(x)) / cos(x). Multiply numerator and denominator by (1 + sin(x)): [(1 - sin(x))(1 + sin(x))] / [cos(x)(1 + sin(x))]. Numerator is a difference of squares: 1 - sin²(x) = cos²(x). The expression becomes: cos²(x) / [cos(x)(1 + sin(x))]. Cancel one cos(x): cos(x) / (1 + sin(x)). This equals the right side.
10. Start with the right side: (tan²(x) - 1) / (tan²(x) + 1). Rewrite tan²(x) as sin²(x)/cos²(x): [ (sin²(x)/cos²(x)) - 1 ] / [ (sin²(x)/cos²(x)) + 1 ]. Multiply numerator and denominator by cos²(x): (sin²(x) - cos²(x)) / (sin²(x) + cos²(x)). Denominator is 1. Numerator: sin²(x) - cos²(x) = (1 - cos²(x)) - cos²(x) = 1 - 2cos²(x). This equals the left side.
11. Start with the right side: csc²(θ)tan²(θ) - 1. Rewrite in terms of sin and cos: (1/sin²(θ)) * (sin²(θ)/cos²(θ)) - 1 = 1/cos²(θ) - 1 = sec²(θ) - 1. Use the Pythagorean identity sec²(θ) - 1 = tan²(θ). This equals the left side.
12. Start with the right side: cos(x) / (1 - sin(x)). Multiply numerator and denominator by (1 + sin(x)): [cos(x)(1 + sin(x))] / [(1 - sin(x))(1 + sin(x))]. Denominator: 1 - sin²(x) = cos²(x). The expression becomes: [cos(x)(1 + sin(x))] / cos²(x) = (1 + sin(x)) / cos(x) = 1/cos(x) + sin(x)/cos(x) = sec(x) + tan(x). This equals the left side.
13. Start with the left side: csc(β)/sin(β) - cot(β)/tan(β). Rewrite in terms of sin and cos: (1/sin(β)) / sin(β) - (cos(β)/sin(β)) / (sin(β)/cos(β)). Simplify each term: 1/sin²(β) - (cos(β)/sin(β)) * (cos(β)/sin(β)) = 1/sin²(β) - cos²(β)/sin²(β). Combine: (1 - cos²(β)) / sin²(β) = sin²(β) / sin²(β) = 1. This equals the right side.
14. Start with the left side: sin⁴(x) - cos⁴(x). Factor as a difference of squares: (sin²(x) - cos²(x))(sin²(x) + cos²(x)) = (sin²(x) - cos²(x)) * 1 = sin²(x) - cos²(x). Rewrite sin²(x) as 1 - cos²(x): (1 - cos²(x)) - cos²(x) = 1 - 2cos²(x). This equals the right side.
15. Start with the left side: (sin(x) - cos(x))² + (sin(x) + cos(x))². Expand both squares: [sin²(x) - 2sin(x)cos(x) + cos²(x)] + [sin²(x) + 2sin(x)cos(x) + cos²(x)]. Combine like terms: sin²(x) + cos²(x) + sin²(x) + cos²(x) = 1 + 1 = 2. This equals the right side.
16. Start with the left side: (sin²(x) + 4sin(x) + 3) / cos²(x). Factor the numerator: (sin(x) + 1)(sin(x) + 3). Denominator: cos²(x) = 1 - sin²(x) = (1 - sin(x))(1 + sin(x)). The expression becomes: [(sin(x) + 1)(sin(x) + 3)] / [(1 - sin(x))(1 + sin(x))]. Cancel (sin(x) + 1): (sin(x) + 3) / (1 - sin(x)). This equals the right side.
17. Start with the left side: cos(x)/(1 - sin(x)) - tan(x). From problem 12, we know cos(x)/(1 - sin(x)) = sec(x) + tan(x). Substitute: (sec(x) + tan(x)) - tan(x) = sec(x). This equals the right side.
18. Start with the left side: tan²(x) + 1 + tan(x)sec(x). Use the identity tan²(x) + 1 = sec²(x): sec²(x) + tan(x)sec(x). Factor out sec(x): sec(x)(sec(x) + tan(x)). From problem 12, we know sec(x) + tan(x) = cos(x)/(1 - sin(x)). So the expression is sec(x) * [cos(x)/(1 - sin(x))]. Since sec(x) = 1/cos(x), this becomes (1/cos(x)) * (cos(x)/(1 - sin(x))) = 1/(1 - sin(x)). Multiply numerator and denominator by (1 + sin(x)): (1 + sin(x)) / [(1 - sin(x))(1 + sin(x))] = (1 + sin(x)) / (1 - sin²(x)) = (1 + sin(x)) / cos²(x). This equals the right side.
2. Start with the left side: 1/tan(x) + tan(x). Rewrite 1/tan(x) as cot(x): cot(x) + tan(x). Express both in terms of sin and cos: cos(x)/sin(x) + sin(x)/cos(x). Combine over a common denominator: (cos²(x) + sin²(x)) / (sin(x)cos(x)). Use the Pythagorean identity: 1 / (sin(x)cos(x)). This equals the right side.
3. Start with the left side: sin(x) - sin(x)cos²(x). Factor out sin(x): sin(x)(1 - cos²(x)). Use the Pythagorean identity 1 - cos²(x) = sin²(x): sin(x) * sin²(x) = sin³(x). This equals the right side.
4. Start with the left side: cos(α)/(1 + sin(α)) + (1 + sin(α))/cos(α). Combine over a common denominator: [cos²(α) + (1 + sin(α))²] / [cos(α)(1 + sin(α))]. Expand the numerator: cos²(α) + 1 + 2sin(α) + sin²(α). Group terms: (cos²(α) + sin²(α)) + 1 + 2sin(α) = 1 + 1 + 2sin(α) = 2 + 2sin(α) = 2(1 + sin(α)). The expression becomes: 2(1 + sin(α)) / [cos(α)(1 + sin(α))]. Cancel (1 + sin(α)): 2/cos(α) = 2sec(α). This equals the right side.
5. Start with the left side: cos(x)/(1 - sin(x)) - cos(x)/(1 + sin(x)). Combine over a common denominator: [cos(x)(1 + sin(x)) - cos(x)(1 - sin(x))] / [(1 - sin(x))(1 + sin(x))]. Factor cos(x) in the numerator: cos(x)[(1 + sin(x)) - (1 - sin(x))] = cos(x)[1 + sin(x) - 1 + sin(x)] = cos(x)[2sin(x)] = 2sin(x)cos(x). The denominator is a difference of squares: 1 - sin²(x) = cos²(x). The expression is: 2sin(x)cos(x) / cos²(x) = 2sin(x)/cos(x) = 2tan(x). This equals the right side.
6. Start with the right side: (csc(x)cos(x)) / (tan(x) + cot(x)). Rewrite all functions in terms of sin and cos: ( (1/sin(x)) * cos(x) ) / ( sin(x)/cos(x) + cos(x)/sin(x) ). Simplify numerator: cos(x)/sin(x). Denominator: (sin²(x) + cos²(x)) / (sin(x)cos(x)) = 1 / (sin(x)cos(x)). The expression becomes: (cos(x)/sin(x)) / (1/(sin(x)cos(x))) = (cos(x)/sin(x)) * (sin(x)cos(x)/1) = cos²(x). This equals the left side.
7. Start with the left side: (sin⁴(x) - cos⁴(x)) / (sin²(x) - cos²(x)). Factor the numerator as a difference of squares: (sin²(x) - cos²(x))(sin²(x) + cos²(x)). The expression becomes: [(sin²(x) - cos²(x))(sin²(x) + cos²(x))] / (sin²(x) - cos²(x)). Cancel (sin²(x) - cos²(x)): sin²(x) + cos²(x) = 1. This equals the right side.
8. Start with the left side: tan²(x) / (tan²(x) + 1). Use the Pythagorean identity tan²(x) + 1 = sec²(x): tan²(x) / sec²(x). Rewrite in terms of sin and cos: (sin²(x)/cos²(x)) / (1/cos²(x)) = (sin²(x)/cos²(x)) * (cos²(x)/1) = sin²(x). This equals the right side.
9. Start with the left side: (1 - sin(x)) / cos(x). Multiply numerator and denominator by (1 + sin(x)): [(1 - sin(x))(1 + sin(x))] / [cos(x)(1 + sin(x))]. Numerator is a difference of squares: 1 - sin²(x) = cos²(x). The expression becomes: cos²(x) / [cos(x)(1 + sin(x))]. Cancel one cos(x): cos(x) / (1 + sin(x)). This equals the right side.
10. Start with the right side: (tan²(x) - 1) / (tan²(x) + 1). Rewrite tan²(x) as sin²(x)/cos²(x): [ (sin²(x)/cos²(x)) - 1 ] / [ (sin²(x)/cos²(x)) + 1 ]. Multiply numerator and denominator by cos²(x): (sin²(x) - cos²(x)) / (sin²(x) + cos²(x)). Denominator is 1. Numerator: sin²(x) - cos²(x) = (1 - cos²(x)) - cos²(x) = 1 - 2cos²(x). This equals the left side.
11. Start with the right side: csc²(θ)tan²(θ) - 1. Rewrite in terms of sin and cos: (1/sin²(θ)) * (sin²(θ)/cos²(θ)) - 1 = 1/cos²(θ) - 1 = sec²(θ) - 1. Use the Pythagorean identity sec²(θ) - 1 = tan²(θ). This equals the left side.
12. Start with the right side: cos(x) / (1 - sin(x)). Multiply numerator and denominator by (1 + sin(x)): [cos(x)(1 + sin(x))] / [(1 - sin(x))(1 + sin(x))]. Denominator: 1 - sin²(x) = cos²(x). The expression becomes: [cos(x)(1 + sin(x))] / cos²(x) = (1 + sin(x)) / cos(x) = 1/cos(x) + sin(x)/cos(x) = sec(x) + tan(x). This equals the left side.
13. Start with the left side: csc(β)/sin(β) - cot(β)/tan(β). Rewrite in terms of sin and cos: (1/sin(β)) / sin(β) - (cos(β)/sin(β)) / (sin(β)/cos(β)). Simplify each term: 1/sin²(β) - (cos(β)/sin(β)) * (cos(β)/sin(β)) = 1/sin²(β) - cos²(β)/sin²(β). Combine: (1 - cos²(β)) / sin²(β) = sin²(β) / sin²(β) = 1. This equals the right side.
14. Start with the left side: sin⁴(x) - cos⁴(x). Factor as a difference of squares: (sin²(x) - cos²(x))(sin²(x) + cos²(x)) = (sin²(x) - cos²(x)) * 1 = sin²(x) - cos²(x). Rewrite sin²(x) as 1 - cos²(x): (1 - cos²(x)) - cos²(x) = 1 - 2cos²(x). This equals the right side.
15. Start with the left side: (sin(x) - cos(x))² + (sin(x) + cos(x))². Expand both squares: [sin²(x) - 2sin(x)cos(x) + cos²(x)] + [sin²(x) + 2sin(x)cos(x) + cos²(x)]. Combine like terms: sin²(x) + cos²(x) + sin²(x) + cos²(x) = 1 + 1 = 2. This equals the right side.
16. Start with the left side: (sin²(x) + 4sin(x) + 3) / cos²(x). Factor the numerator: (sin(x) + 1)(sin(x) + 3). Denominator: cos²(x) = 1 - sin²(x) = (1 - sin(x))(1 + sin(x)). The expression becomes: [(sin(x) + 1)(sin(x) + 3)] / [(1 - sin(x))(1 + sin(x))]. Cancel (sin(x) + 1): (sin(x) + 3) / (1 - sin(x)). This equals the right side.
17. Start with the left side: cos(x)/(1 - sin(x)) - tan(x). From problem 12, we know cos(x)/(1 - sin(x)) = sec(x) + tan(x). Substitute: (sec(x) + tan(x)) - tan(x) = sec(x). This equals the right side.
18. Start with the left side: tan²(x) + 1 + tan(x)sec(x). Use the identity tan²(x) + 1 = sec²(x): sec²(x) + tan(x)sec(x). Factor out sec(x): sec(x)(sec(x) + tan(x)). From problem 12, we know sec(x) + tan(x) = cos(x)/(1 - sin(x)). So the expression is sec(x) * [cos(x)/(1 - sin(x))]. Since sec(x) = 1/cos(x), this becomes (1/cos(x)) * (cos(x)/(1 - sin(x))) = 1/(1 - sin(x)). Multiply numerator and denominator by (1 + sin(x)): (1 + sin(x)) / [(1 - sin(x))(1 + sin(x))] = (1 + sin(x)) / (1 - sin²(x)) = (1 + sin(x)) / cos²(x). This equals the right side.
Parent Tip: Review the logic above to help your child master the concept of trig identities worksheet.
