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Improper Integrals

Express sin 1...

Question

Express sin 10 theta in terms of functions of 5 theta

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Solution

$\sin 10 θ = \sin 2 (5 θ) = 2 \sin 5 θ \cos 5 θ$

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Q. Prove :
Express

s i n 5 θ

s i n θ

and hence find the value of

s i n 36^{\circ}

tan θ

sin θ

Q. Question 5 (iii)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(iii)

\frac{t a n θ}{(1 - c o t θ)} + \frac{c o t θ}{(1 - t a n θ)} = 1 + s e c θ c o s e c θ

[Hint : Write the expression in terms of

s i n θ

c o s θ

]

Q. Question 5 (iii)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
(iii)

\frac{t a n θ}{(1 - c o t θ)} + \frac{c o t θ}{(1 - t a n θ)} = 1 + s e c θ c o s e c θ

[Hint : Write the expression in terms of

s i n θ

c o s θ

sin (θ + α) = cos (θ + α)

tan θ

α

.
(ii) Find the value of

tan (π / 4 + θ) . tan (π / 4 - θ)

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