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Factoring Trigonometric IdentitiesFactoring Trigonometric Identities
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Question 1 of 2
1. Question
Factorise`2sec^2A-3secA+1`Hint
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The cross method is a factorisation method used for quadratics.Since the equation is in standard form `(``a``x^2+``b``x+``c``=0)` we can factorise using the cross method.`2\text(sec)^2A` `-3\text(sec)A` `+1``=0`To factorise, we need to find two values on the left side that multiply to `2\text(sec)^2A` and two values on the right side that multiply to `1` and, when cross multiplied with the left side values and added together, gives `-3\text(sec)A`For the left side, `2\text(sec)A` and `\text(sec)A` fit the condition`2\text(sec)A xx \text(sec)A` `=` `2\text(sec)^2A` For the right side, `-1` and `-1` fit both conditions`[2\text(sec)Axx(-1)]+[\text(sec)Axx(-1)]` `=` `-3\text(sec)A` `-1 xx -1` `=` `1` 
Read across to get the factors.`(2\text(sec)-1)(\text(sec)-1)``(2\text(sec)-1)(\text(sec)-1)` -
Question 2 of 2
2. Question
Factorise`2sin^2 theta+7sin theta cos theta +3cos^2 theta`Hint
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The cross method is a factorisation method used for quadratics.Since the equation is in standard form `(``a``x^2+``b``x+``c``=0)` we can factorise using the cross method.`2 \text(sin)^2theta` `+7 \text(sin) theta \text(cos) theta` `+3 \text(cos)^2theta``=0`To factorise, we need to find two values on the left side that multiply to `2 \text(sin)^2theta` and two values on the right side that multiply to `3 \text(cos)^2theta` and, when cross multiplied with the left side values and added together, gives `7 \text(sin) theta \text(cos) theta`For the left side, `2 \text(sin) theta` and `\text(sin) theta` fit the condition`2 \text(sin) theta xx \text(sin) theta` `=` `2 \text(sin)^2theta` For the right side, `\text(cos) theta` and `3 \text(cos) theta` fit both conditions`(\text(cos) thetaxx\text(sin) theta)+(3 \text(cos) thetaxx2 \text(sin) theta)` `=` `7 \text(sin) theta \text(cos) theta` `\text(cos) theta xx 3 \text(cos) theta` `=` `3 \text(cos)^2theta` Read across to get the factors.`(2 \text(sin) theta+\text(cos) theta)(\text(sin) theta+3 \text(cos) theta)``(2 \text(sin) theta+\text(cos) theta)(\text(sin) theta+3 \text(cos) theta)`