QQuestion
If sin x = m sin y and tan x = n tan y, then show that cos² x = (m² - 1)\over (n² - 1)
✓Answer
Given: sin x = m sin y and tan x = n tan y
We know,
tan x = sin x \over cos x
and tan y = sin y \over cos y
So,
n = tan x \over tan y
= sin x \over cos x ÷ sin y \over cos y
= sin x \over sin y × cos y \over cos x
Substitute sin x \over sin y = m
⇒ n = m × cos y \over cos x
So, cos x \over cos y = m \over n
⇒ cos²x \over cos²y = m² \over n²
⇒ cos²x = m² \over n² × cos²y —– (1)
Now, from sin x = m sin y,
sin²x = m² sin²y
⇒ 1 – cos²x = m²(1 – cos²y)
Substitute (1):
1 – (m² / n²)cos²y = m² – m²cos²y
⇒ 1 – m²/n² cos²y = m² – m²cos²y
⇒ n² – m²cos²y = n²m² – n²m²cos²y
⇒ n² – n²m² = cos²y (m²n² – m²)
⇒ cos²y = (n² – 1) / (n² – m²)
cos²x = (m² / n²) × cos²y
= (m² / n²) × (n² – 1)/(n² – m²)
= (m² – 1)/(n² – 1)
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