7.1.
This formula is the source of a variety of other important formulas in trigonometry.

7.2.
Some other formulas can also be selected to be a queen. Examples are sums and differences formulas (7.8) – (7.11).

7.3.

7.4.

7.5.

7.6.
1) cos 7A·cos3A + sin 7A·sin 3A = cos (7A – 3A) = cos 4A.

2) sin5x·cos3x – cos5x·sin3x = sin(5x – 3x) = sin 2x

7.7.
cos · cos (17° –) – sin· sin (17° –) = cos [+ (17° – )] = cos 17°

7.8.
By formulas (7.19) and (7.20), 1) 2sin 40°cos 20°; 2) –2sin 5· sin 3

7.9.
By formulas (7.12) – (7.14),
1) ½( sin 50°+ sin 10°);
2) ½( cos 12°– cos 68°);
3) ½( cos 8 + cos 2)

7.10.
In formulas (7.30) and (7.31), replace with 2.

7.11.
1) tan( + ) = sin( + )/ cos( + ). Apply formulas (7.11) and (7.9). Then divide both parts of the fraction by cos·cos .

2) Apply formulas (7.12) and (7.29).

3) Using formula (7.26), write the right part as tan ·sin2 = (sin/cos)·2sin ·cos= 2sin². Then apply formula (7.28).

7.12
Use the formula (7.29) twice: cos 4 = 2cos² 2– 1 and cos 2 = 2cos²– 1. The answer is: cos 4 = 8cos⁴– 8cos² +1.

7.13.
Multiply and divide left part by sin, and apply formula (7.26) n times.