Is sin(2a)+cos(a)=cos(a)(2sin(a)+1) ?

The answer to whether sin(2a)+cos(a)=cos(a)(2sin(a)+1) is True

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prove sin(2a)+cos(a)=cos(a)(2sin(a)+1)

prove

sin (2 a) + cos (a) = cos (a) (2 sin (a) + 1)

True

Solution steps

sin (2 a) + cos (a) = cos (a) (2 sin (a) + 1)

Manipulating left side

sin (2 a) + cos (a)

Rewrite using trig identities

sin (2 a) + cos (a)

Use the Double Angle identity:

sin (2 x) = 2 sin (x) cos (x)

= 2 sin (a) cos (a) + cos (a)

= 2 sin (a) cos (a) + cos (a)

Manipulating right side

cos (a) (2 sin (a) + 1)

Expand

(1 + 2 sin (a)) cos (a) : cos (a) + 2 cos (a) sin (a)

(1 + 2 sin (a)) cos (a)

= cos (a) (1 + 2 sin (a))

Apply the distributive law:

a (b + c) = ab + a c

a = cos (a), b = 1, c = 2 sin (a)

= cos (a) \cdot 1 + cos (a) \cdot 2 sin (a)

= 1 \cdot cos (a) + 2 cos (a) sin (a)

Multiply:

1 \cdot cos (a) = cos (a)

= cos (a) + 2 cos (a) sin (a)

= cos (a) + 2 cos (a) sin (a)

We showed that the two sides could take the same form

\Rightarrow True

\frac{csc ( x )}{2 cos ( x )} = csc (2 x)

tan^{2} (u) = \frac{1 - cos ( 2 u )}{1 + cos ( 2 u )}

\frac{1}{sin ( x ) - 1} - \frac{1}{sin ( x ) + 1} = - 2 sec^{2} (x)

(1 + sin (θ)) (1 - sin (θ)) = \frac{1}{sec ^{2} ( θ )}

csc^{4} (x) - csc^{2} (x) = cot^{2} (+ cot^{4} (x))

Is sin(2a)+cos(a)=cos(a)(2sin(a)+1) ?
The answer to whether sin(2a)+cos(a)=cos(a)(2sin(a)+1) is True