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Trigonometric Identities Worksheet

Given: Fundamental Identities

sin cos 1 csc = sin 2 sin + cos 2 = 1 tan =

cos sin 1 sec = cos 2 tan + 1 = sec 2 cot =

1 tan 2 1 + cot = csc 2 cot =

Establish (or verify) the following identities. 1.

1 ­ tan cot ­ 1 = 1 + tan cot + 1

2.

cos2 x = 1 ­ sin x 1 + sin x

3.

tan sin = sec ­ cos

4.

sin 1 + cos + = 2 csc 1 + cos sin

5.

4 If tan = ­ , 3

1 < < and cos = ­ , 2 2

< <

3 , find the exact value of: 2

(a) sin ( + )

(b) tan ( + )

6.

Use the given angle sum and difference identities to verify the following identities: cos (A + B) = cos A cos B ­ sin A sin B and cos (A ­ B) = cos A cos B + sin A sin B

(a) cos ­ = sin 2

(b) cos ( + ) = ­cos

7.

Use the angle sum identity sin (A + B) = sin A cos B + cos A sin B and sin (A ­ B) = sin A cos B ­ cos A sin B to verify the following identities:

(a) sin ­ = cos 2

(b) sin ( ­ ) = sin

8.

Use one of the Half-Angle formulas to find the exact value of (a) cos 15°

(b) tan 12

9.

If cos = (a) sin ( 2 )

4 and 0 < < , find the exact value of 5 2

(b) cos ( 2 )

(c) the quadrant in which 2 lies

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