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Trigonometric Identities

Question 5 ix...

Question

Question 5 (ix)
Prove the following identities, where the angles involved are acute angles for which the expressions are defined.
$(i x) (c o s e c A - s i n A) (s e c A - c o s A) = \frac{1}{(t a n A + c o t A)}$
[Hint : Simplify L.H.S and R.H.S separately]

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Solution

$(i x) (c o s e c A - s i n A) (s e c A - c o s A) = \frac{1}{(t a n A + c o t A)}$
$L . H . S = (c o s e c A - s i n A) (s e c A - c o s A) = (\frac{1}{s i n A} - s i n A) (\frac{1}{c o s A} - c o s A) = [\frac{(1 - s i n^{2} A)}{s i n A}] [\frac{(1 - c o s^{2} A)}{c o s A}] = (\frac{c o s^{2} A}{s i n A}) \times (\frac{s i n^{2} A}{c o s A}) = c o s A s i n A R . H . S = \frac{1}{(t a n A + c o t A)} = \frac{1}{(\frac{s i n A}{c o s A} + \frac{c o s A}{s i n A})} = \frac{1}{[\frac{(s i n^{2} A + c o s^{2} A)}{s i n A c o s A}]} = c o s A s i n A L . H . S = R . H . S$

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Similar questions

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

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

[Hint : Simplify L.H.S and R.H.S separately]

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

\frac{(1 + s e c A)}{s e c A} = \frac{s i n^{2} A}{(1 - c o s A)}

[Hint : Simplify L.H.S and R.H.S separately]

Q. Prove the following identities, where the angles involved are acute angles for which the expressions are defined.

\frac{(1 + s e c A)}{s e c A} = \frac{s i n^{2} A}{(1 - c o s A)}

[Hint : Simplify L.H.S and R.H.S separately]

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

\frac{(1 + s e c A)}{s e c A} = \frac{s i n^{2} A}{(1 - c o s A)}

[Hint : Simplify L.H.S and R.H.S separately]

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

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

, using the identity

c o s e c^{2} A = 1 + c o t^{2} A

.

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Trigonometric Identities

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