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Math Equations

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1.
sin^4 (x) + cos^4 (x)
= (sin^2 x)^2 + (cos^2 x)^2 + 2 . sin^2 x . cos^2 x - 2 . sin^2 x . cos^2 x
= (sin^2 x + cos^2 x)^2 - 2 . sin^2 x . cos^2 x
= 1 - 2 . sin^2 x . cos^2 x
= 1 - 2 . sin^2 x . (1 - sin^2 x )
= 1 - 2 . sin^2 x + 2 sin^4 x --Answer(4).

2.
(sin(a+b) + cos(a-b))/(cos(a+b) - sin(a-b))
= (sina.cosb + cosa.sinb + cosa.cosb + sina.sinb)/(cosa.cosb - sina.sinb - sina.cosb + cosa.sinb)
= (sina.(cosb+sinb) + cosa.(cosb+sinb))/(cosa.(cosb+sinb) - sina.(cosb+sinb))
= (sina +cosa).(cosb+sinb)/((cosa-sina)(cosb+sinb))
= (sina + cosa)/(cosa - sina) --Answer(1).

3.
a == alpha, b == beta
cos^2(a+b) - cos^2(a-b)
= (cos(a+b) + cos(a-b)) . (cos(a+b) - cos(a-b))
= (cosa . cosb - sina . sinb + cosa . cosb + sina. sinb ) . (cosa . cosb - sina . sinb - cosa.cosb - sina.sinb)
= 2.cosa.cosb . (-2sina.sinb)
= -4.sina.cosa.sinb.cosb --Answer(1)

4.
tana +cot(2a) = tana + 1/tan(2a)
=tana + cos(2a)/sin(2a)
= sina/cosa + (cos^2 a - sin^2 a)/(2.sina.cosa)
= (sina.2.sina + cos^2 a - sin^2 a)/(2sina.cosa)
= (2.sin^2 a + cos^2 a - sin^2 a)/sin(2a)
= (sin^2 a + cos^2 a)/sin(2a)
= 1/sin(2a) --Answer(4)

5.
(tan(2a) + sin(2a))/tan(2a)
= (sin(2a)/cos(2a) + sin(2a))/(sin(2a)/cos(2a))
= ((sin2a + sin(2a).cos(2a))/cos(2a))/(sin(2a)/cos(2a))
= sin(2a).(1 + cos(2a))/sin(2a)
= 1 + cos(2a)
= 1 + 2cos^2 a - 1
= 2.cos^2 a --Answer(3)

6.
tan(45+a) - tan(45-a)
= (1+ tana)/(1- tana) - (1-tana)/(1+tana)
= ((1+tana)^2 - (1-tana)^2)/(1 - tan^2 a)
= ...

Solution Summary

A few trigonometry equation problems are solved here.

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