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In this lesson, we will learn how to derive trigonometric addition formulae and use them to simplify and evaluate trigonometric expressions.

Q1:

Simplify c o s c o s s i n s i n 2 π 2 2 π β 2 π 2 2 π .

Q2:

Simplify c o s c o s s i n s i n 4 1 π 7 π β 4 1 π 7 π .

Q3:

Using the relation t a n t a n t a n t a n t a n ( πΌ + π½ ) = πΌ + π½ 1 β πΌ π½ , find an expression for t a n ( πΌ β π½ ) in terms of t a n πΌ and t a n π½ which holds when ( πΌ β π½ ) β π 2 + π π .

Q4:

Simplify s i n c o s c o s s i n ( 4 8 π + 4 2 π ) 4 2 π β ( 4 8 π + 4 2 π ) 4 2 π .

Q5:

Simplify s i n c o s c o s s i n 1 4 7 1 2 0 β 1 4 7 1 2 0 β β β β .

Q6:

Simplify s i n c o s c o s s i n 1 1 7 1 5 4 β 1 1 7 1 5 4 β β β β .

Q7:

Simplify s i n c o s c o s s i n 1 1 5 1 6 4 β 1 1 5 1 6 4 β β β β .

Q8:

Simplify s i n c o s c o s s i n 4 0 1 0 2 β 4 0 1 0 2 β β β β .

Q9:

Simplify s i n c o s c o s s i n 1 3 2 2 7 β 1 3 2 2 7 β β β β .

Q10:

Simplify s i n c o s c o s s i n 5 6 1 4 1 β 5 6 1 4 1 β β β β .

Q11:

Simplify c o s c o s s i n s i n ( 2 3 π + 2 5 π ) 2 5 π + ( 2 3 π + 2 5 π ) 2 5 π .

Q12:

In the given figure, π π π π is a rectangle and the length of π π is 1.

Find the lengths of π π and π π in terms of πΌ and π½ .

Find π in terms of πΌ and π½ . Hence find the lengths of π π and π π .

By considering a suitable angle, find the lengths of π π and π π .

Use your answers to the previous parts of the question to find expressions for s i n ( πΌ β π½ ) and c o s ( πΌ β π½ ) .

Q13:

In the figure, which triangles are similar?

Given that π΅ πΆ = 1 , find expressions for the lengths of π΄ πΆ , πΆ π· , π΄ π· , and πΆ πΉ .

Find an expression for t a n ( πΌ β π½ ) .

Q14:

Consider the given figure.

Find the lengths π΄ and π΅ in terms of πΌ and π½ .

Find the lengths πΆ , π· , πΈ , and πΉ in terms of πΌ and π½ .

By writing π΄ and π΅ in terms of πΆ , π· , πΈ , and πΉ , find expressions for c o s ( πΌ + π½ ) and s i n ( πΌ + π½ ) in terms of c o s πΌ , c o s π½ , s i n πΌ , and s i n π½ .

Q15:

Using the relation t a n s i n c o s π = π π , find an expression for t a n ( πΌ + π½ ) in terms of t a n πΌ and t a n π½ which holds when ( πΌ + π½ ) β π 2 + π π .

Q16:

Simplify t a n t a n t a n t a n 1 5 9 β 1 1 4 1 + 1 5 9 1 1 4 β β β β .

Q17:

Simplify t a n t a n t a n t a n 1 5 6 β 4 3 1 + 1 5 6 4 3 β β β β .

Q18:

Simplify t a n t a n t a n t a n 1 6 7 β 1 0 2 1 + 1 6 7 1 0 2 β β β β .

Q19:

Simplify t a n t a n t a n t a n 2 2 β 1 6 1 + 2 2 1 6 β β β β .

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