If Tn-sin n-cos n x- prove thati T3-T5T1-T5-T7T3ii 2 T 6-3 T 4-1-0iii 6 T 10-15 T 8-10 T 6-1-0

(i) LHS: T3 T5T1=sin3x+cos3x sin5x+cos5xsinx+cosx= #160;sin3x sin5x+cos3x cos5xsinx+cosx=sin3x1 sin2x+cos3x1 cos2xsinx+cosx=sin3x.cos2x+cos3x.sin2xsinx+cosx= ...

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Byju's Answer

Standard XII

Mathematics

Properties Derived from Trigonometric Identities

If Tn=sin n+c...

Question

If $T_{n} = \sin^{n} x + \cos^{n} x$ , prove that

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

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

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

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Solution

(i) LHS:

$\frac{T_{3} - T_{5}}{T_{1}} = \frac{(\sin^{3} x + \cos^{3} x) - (\sin^{5} x + \cos^{5} x)}{\sin x + \cos x} = \frac{\sin^{3} x - \sin^{5} x + \cos^{3} x - \cos^{5} x}{\sin x + \cos x} = \frac{\sin^{3} x (1 - \sin^{2} x) + \cos^{3} x (1 - \cos^{2} x)}{\sin x + \cos x} = \frac{\sin^{3} x . \cos^{2} x + c {os}^{3} x . \sin^{2} x}{\sin x + \cos x} = \frac{\sin^{2} x . \cos^{2} x (\sin x + c os x)}{\sin x + \cos x} = \sin^{2} x . \cos^{2} x$

RHS:
$\frac{T_{5} - T_{7}}{T_{3}} = \frac{(\sin^{5} x + \cos^{5} x) - (\sin^{7} x + \cos^{7} x)}{\sin^{3} x + \cos^{3} x} = \frac{\sin^{5} x - si n^{7} x + \cos^{5} x - \cos^{7} x}{\sin^{3} x + \cos^{3} x} = \frac{\sin^{5} x (1 - \sin^{2} x) + \cos^{5} x (1 - \cos^{2} x)}{\sin^{3} x + \cos^{3} x} = \frac{\sin^{5} x \cos^{2} x + \cos^{5} x \sin^{2} x}{\sin^{3} x + \cos^{3} x} = \sin^{2} x . \cos^{2} x$

LHS = RHS

Hence proved.

(ii) LHS:

$2 T_{6} - 3 T_{4} + 1 2 (\sin^{6} x + \cos^{6} x) - 3 (\sin^{4} x + \cos^{4} x) + 1 2 (\sin^{2} x + \cos^{2} x) (\sin^{4} x + \cos^{4} x - \sin^{2} x \cos^{2} x) - 3 (\sin^{4} x + \cos^{4} x) + 1 2.1 . (\sin^{4} x + \cos^{4} x - \sin^{2} x \cos^{2} x) - 3 (\sin^{4} x + \cos^{4} x) + 1 2 \sin^{4} x + 2 \cos^{4} x - 2 \sin^{2} x \cos^{2} x - 3 \sin^{4} x - 3 \cos^{4} x + 1 - (\sin^{4} x + \cos^{4} x) - \sin^{2} x \cos^{2} x + 1 - (\sin^{2} x + \cos^{2} x)^{2} + 1 - 1 + 1 0$

Hence proved.

(iii) LHS:

$6 T_{10} - 15 T_{8} + 10 T_{6} - 1 6 (\sin^{10} x + \cos^{10} x) - 15 (\sin^{8} x + \cos^{8} x) + 10 (\sin^{6} x + \cos^{6} x) - 1$

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Similar questions

Q. If

T_{n} = \sin^{n} θ + \cos^{n} θ

, prove that

(i)

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

(ii)

2 T_{6} - 3 T_{4} + 1 = 0

(iii)

6 T_{10} - 15 T_{8} + 10 T_{6} - 1 = 0

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
(i)

2 T_{6} - 3 T_{4} + 1 = 0

(ii)

6 T_{10} - 15 T_{8} + 10 T_{6} - 1 = 0

Q. If

T_{n} = {sin}^{n} x + {cos}^{n} x

, then find the value of

6 T_{10} - 15 T_{8} + 10 T_{6}

Q. If

T_{n} = {sin}^{n} x + {cos}^{n} x

, then find the value of

2 T_{6} - 3 T_{4} + 10.

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If Tn=sinn x+cosn x, prove that (i) T3-T5T1=T5-T7T3 (ii) 2 T6-3 T4+1=0 (iii) 6T10-15 T8+10 T6-1=0

If $T_{n} = \sin^{n} x + \cos^{n} x$ , prove that

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

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

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