Tan(a±b) and Double Angle

Tan(a±b) and Double Angle

What is it?

tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined). tan(A-B)=tan A - tan B1 + tan A tan B. Double: sin 2A = 2 sin A cos A, cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = 2 tan A1-tan² A. Meaning: divide sin compound by cos compound and divide through by cos A cos B.

In simple words

• tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined).
• tan(A-B)=tan A - tan B1 + tan A tan B.
• Double: sin 2A = 2 sin A cos A, cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = 2 tan A1-tan² A.
• Bottom 1 - tan A tan B. If you write 1 + by accident, A+B is wrong.

Important terms

TAN(A±B) AND DOUBLE ANGLE: tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined). tan(A-B)=tan A - tan B1 + tan A tan B. Double: sin 2A = 2 sin A cos A, cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = 2 tan A1-tan² A. Meaning: divide sin compound by cos compound and divide through by cos A cos B.

1. TAN SUM: Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).

2. DOUBLE: Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

What you should learn

• State the main ideas and notation used in tan(a±b) and double angle.
• Apply the rules of tan(a±b) and double angle in a clear, step-by-step solution.
• Check signs, conditions, and the final answer for mathematical accuracy.

Examples

• 1 - product in bottom
• Defined when cos(A+B)!=0
• tan(A-B) plus in bottom

TAN SUM

Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).

DOUBLE

Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

Full school-level explanation

tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined). tan(A-B)=tan A - tan B1 + tan A tan B. Double: sin 2A = 2 sin A cos A, cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = 2 tan A1-tan² A. Meaning: divide sin compound by cos compound and divide through by cos A cos B.

Divide sin/cos, then double:

WHAT — TAN(A±B) AND DOUBLE ANGLE

tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined). tan(A-B)=tan A - tan B1 + tan A tan B. Double: sin 2A = 2 sin A cos A, cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A, tan 2A = 2 tan A1-tan² A. Meaning: divide sin compound by cos compound and divide through by cos A cos B.

RULE — 1. TAN SUM

Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).

METHOD — 2. DOUBLE

Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

WORKED EXAMPLE

Worked: tan 75 = tan(45+30). tan45=1, tan30=1/sqrt(3) tan75 = 1 + 1/sqrt(3)1 - 1/sqrt(3) times sqrt(3)/sqrt(3): sqrt(3)+1sqrt(3)-1

TAN SUM

Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined when that is 0 (A+B odd multiple of 90 degrees school).

Remember

• divide by cos cos

Examples

• 1 - product in bottom
• Defined when cos(A+B)!=0
• tan(A-B) plus in bottom

DOUBLE

Put B=A. Three forms of cos 2A are all useful (pick the one that matches sin or cos on the paper). Half-angle later if asked via rearranging cos 2A.

Remember

• B=A

Examples

• 2 sin cos
• cos^2-sin^2
• 2t/(1-t^2) for tan

How to answer a tan(a±b) and double angle question

How it works

• Identify the given quantities, notation, and required result.
• Choose the matching rule and write it before substitution.
• Work one algebraic step at a time, keeping brackets and signs clear.
• Check restrictions, simplify the result, and verify it where possible.

Remember

• tan(A+B)=s+t1-st
• sin 2A = 2 sin A cos A
• sqrt(3)+1sqrt(3)-1
• minus in tan(A+B) bottom
• divide by cos cos
• B=A
• tan(A+B)=s+t1-st
• sin 2A = 2 sin A cos A

How the ideas connect

Worked: tan 75 = tan(45+30). tan45=1, tan30=1/sqrt(3) tan75 = 1 + 1/sqrt(3)1 - 1/sqrt(3) times sqrt(3)/sqrt(3): sqrt(3)+1sqrt(3)-1

Deeper understanding

Go beyond memorising Tan(a±b) and Double Angle. Connect the definition to reasons, relationships, exceptions, and exam wording.

How the ideas connect

Worked: tan 75 = tan(45+30). tan45=1, tan30=1/sqrt(3) tan75 = 1 + 1/sqrt(3)1 - 1/sqrt(3) times sqrt(3)/sqrt(3): sqrt(3)+1sqrt(3)-1

Common mistakes

Exam focus

Core Idea

1. TAN SUM

2. DOUBLE

Worked Example

Exam focus

Quick revision

• tan(A+B)=s+t1-st
• sin 2A = 2 sin A cos A
• sqrt(3)+1sqrt(3)-1
• minus in tan(A+B) bottom
• TAN(A±B) AND DOUBLE ANGLE: tan(A+B)=tan A + tan B1 - tan A tan B, provided the cosine in the bottom is not zero (defined).
• 1. TAN SUM: Start from sin(A+B)/cos(A+B). Divide num and den by cos A cos B. Bottom 1 - tan A tan B. Undefined w

Bottom 1 - tan A tan B. If you write 1 + by accident, A+B is wrong.

For long answers on Tan(a±b) and Double Angle: define clearly → explain the main parts → give one example → state one exam caution.