cos3A is equal to:
- Acos
3
A−3sin
2
AcosA - Bcos
3
A+4sin
2
AcosA - Ccos
3
A+3sin
2
AcosA - Dcos
3
A−4sin
2
AcosA
Solution & Step-by-step Explanation
Let's recall the standard triple-angle formula for cosine:
cos3A=4cos
3
A−3cosA
Let's check how this can be related to the options by substituting cos
2
A=1−sin
2
A:
We can write:
4cos
3
A−3cosA=cos
3
A+3cos
3
A−3cosA
=cos
3
A+3cosA(cos
2
A−1)
Since cos
2
A−1=−sin
2
A:
=cos
3
A+3cosA(−sin
2
A)
=cos
3
A−3sin
2
AcosA
Thus, option A is identical to cos3A.
cos3A=4cos
3
A−3cosA
Let's check how this can be related to the options by substituting cos
2
A=1−sin
2
A:
We can write:
4cos
3
A−3cosA=cos
3
A+3cos
3
A−3cosA
=cos
3
A+3cosA(cos
2
A−1)
Since cos
2
A−1=−sin
2
A:
=cos
3
A+3cosA(−sin
2
A)
=cos
3
A−3sin
2
AcosA
Thus, option A is identical to cos3A.