If T T -cos n - prove thati 3-t5-T5-T5 T 3-T3

1) T_3 t_5/T_1 = T_5 T_7/T_3 LHS sin^3 θ + cos^3 θ sin^5 θ cos^5 θ/sin θ + cos θ =sin^3 θ + sin^5 θ cos^5 θ/sin θ + cos θ =sin^3 θ(1 sin^2 θ)+cos^3 θ (1 c ...

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Byju's Answer

Standard XII

Physics

Basic Differentiation Rule

If T T θ+cos ...

Question

If $T_{n} θ + c o s^{n} θ$ , prove that
i) $\frac{T_{3} - t_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}$

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Solution

1) $\frac{T_{3} - t_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}$
LHS $\frac{s i n^{3} θ + c o s^{3} θ - s i n^{5} θ - c o s^{5} θ}{s i n θ + c o s θ}$
$= \frac{s i n^{3} θ + s i n^{5} θ - c o s^{5} θ}{s i n θ + c o s θ}$
$= \frac{s i n^{3} θ (1 - s i n^{2} θ) + c o s^{3} θ (1 - c o s^{2} θ)}{s i n θ + c o s θ}$
$= \frac{s i n^{3} θ . c o s^{3} θ + c o s^{3} θ . s i n^{2} θ}{s i n θ + c o s θ}$
$= s i n^{2} θ . c o s^{2} θ$
RHS $\frac{s i n^{5} θ = s i n^{7} s i n θ + c o s^{5} θ - c o s^{7} θ}{s i n^{3} θ + c o s^{3} θ}$
$= \frac{s i n^{5} θ (1 - s i n^{2} θ) + c o s^{5} θ (1 - c o s θ)}{s i n^{3} θ + c o s^{3} θ}$
$= \frac{s i n^{5} θ . c o s^{2} θ + c o s^{5} θ . s i n^{2} θ}{(s i n^{3} θ + c o s^{3} θ)}$ (Now take out common terms)
$= s i n^{2} θ . c o s^{2} θ$ =LHS proved

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Similar questions

If $T_{n} = s i n^{n} θ + c o s^{n} θ$ , prove that

$(i) \frac{T_{3} - T_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}$

$(i i) 2 T_{6} - 3 T_{4} + 1 = 0$

$(i i i) 6 T_{10} - 15 T_{8} + 10 T_{6} - 1 = 0$

Q. If

T_{n} = {sin}^{n} θ + {cos}^{n} θ

, prove that

\frac{T_{3} - T_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}

Q. If

T_{n} = s i n^{n} θ + c o s^{n} θ

then show that

\frac{T_{3} - T_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}

Q. If

T_{n} = {sin}^{n} θ + {cos}^{n} θ

, then

\frac{_{3} - T_{5}}{T_{1}} \neq \frac{T_{5} - T_{7}}{T_{3}}

Q. Prove the following trigonometric identities.

If T_{n} = \sin^{n} θ + \cos^{n} θ, prove that \frac{T_{3} - T_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}

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If Tnθ+cosnθ, prove that i) T3−t5T1=T5−T7T3

If $T_{n} θ + c o s^{n} θ$ , prove that
i) $\frac{T_{3} - t_{5}}{T_{1}} = \frac{T_{5} - T_{7}}{T_{3}}$