Question

prove that.....(trignometry)

Answer 1

For trigonometry proofs, Learn all trigonometric identities and some of the most used algebraic identities. it will make the proofs a lot easier.

Here, you need the algebraic identity of (a+b)^3.

${(a + b)}^{3} = {a}^{3} + {b}^{3} + 3ab(a + b) \\ \\ \Rightarrow {a}^{3} + {b}^{3} = {(a + b)}^{3} - 3ab(a + b)$

Now, just assume sin^2 A as a and cos^2 A as b. Proof will be done.
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$\text{LHS} = sin^6 A + cos^6 A\\ \\ = (sin^2A)^3+ (cos^2A)^3\\ \\ = [sin^2A+cos^2A]^3 - 3 sin^2A \: cos^2A(sin^2A+cos^2A)\\ \\ = [ 1 ]^3 -3 sin^2A \: cos^2A( 1 )\\ \\ = 1 - 3 sin^2Acos^2A\\ \\ = \text{RHS}$

Answer 2

Here is your answer....

See the attachment..

Hope it helps!!!