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Complementary Trigonometric Ratios

Prove the fol...

Question

Prove the following trigonometric identities.

If cosec θ − sin θ = a³, sec θ − cos θ = b³, prove that a² b² (a² + b²) = 1

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Solution

Given that,

We have to prove

We know that

Now from the first equation, we have

Again from the second equation, we have

Therefore, we have

Hence proved.

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Q. If (cosec θ − sin θ) = a³and (sec θ − cos θ) = b³, prove that

a^{2} b^{2} (a^{2} + b^{2}) = 1 .

Q. Prove the following trigonometric identities.

If cosec θ + cot θ = m and cosec θ − cot θ = n, prove that mn = 1

Prove the following trigonometric identities.
(i) $\frac{1 + cos θ + sin θ}{1 + cos θ - sin θ} = \frac{1 + sin θ}{cos θ}$

(ii) $\frac{sin θ - cos θ + 1}{sin θ + cos θ - 1} = \frac{1}{sec θ - tan θ}$

(iii) $\frac{cos θ - sin θ + 1}{cos θ + sin θ - 1} = cosec θ + cot θ$

(iv) $(sin θ + cos θ) (tan θ + cot θ) = sec θ + cosec θ$

c o s e c θ - s i n θ = a^{3}

s e c θ - c o s θ = b^{3}

then, the value of

a^{2} b^{2} (a^{2} + b^{2})

is __________ .

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