If tan θ = p/q, show that (p sin θ - q cos θ/ p sinθ + q cos θ) = p² - q² / p²+q²
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(p sin θ - q cos θ)/(p sin θ + q cos θ)
= {p(sinθ/cosθ) - q(cos θ/cos θ)}/ {p(sinθ/cosθ)+ q(cosθ/cosθ)}
= (p tan θ -q) / (p tan θ +q)
= {p. (p/q)-q}/{p.(p/q-q)}
= {(p²-q²)/q}/{(p²+q²)/q)
= {(p²-q²)/q}÷{(p²+q²)/q}
= {(p²-q²)/q} × {(q)/(p²+q²)}
= (p²-q²)/(p²+q²)
Hence, L.H.S. = R.H.S.(is roved)
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